SAT Preparation Classes

Quadratic Equations: Factoring

In the last lesson we indicated that there are three ways that quadratic equations can be solved: by factoring, by using the quadratic equation, and by completing the square. It is important to have a thorough understanding of how to factor an expression when a quadratic equation is solved by factoring.
This lesson will concentrate on factoring an expression.

1. A factor of a number is defined as a positive integer that can be divided evenly unto the number – without a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
2. A variable can also be a factor. In the expression 12xy, x is a factor and y is a factor.
3. Negative numbers can also be factors. -1, -2, -3, -4, -6, and -12 are factors of 12.

We may be asked to factor a binomial or a trinomial. To factor a binomial, we factor out the greatest common factor. For example,
Factor 4x + 16.         2 is a common factor but 4 is the greatest common factor. Thus, 4x + 16 = 4(x + 4).
Factor 9a² – 21a.     3a is the greatest common factor. Thus, 9a² – 21a = 3a(3a - 7).

To factor a trinomial into the product of two binomials, we write the expression in the standard format:
ax² + bx + c, and then we find factors that meet two criteria when the coefficient of the x² term is 1:

1. The product of the factors = the last term
2. The sum of the factors = the coefficient of the x term

For example, factor x² + 5x + 4.                x² + 5x + 4 = (x + ?)(x + ?) In place of the question marks, try various factor pairs so that the product of two factors = 4 and the sum of the factors = 5. The factor pairs to consider are 2, 2 and 1, 4. (2)(2) = 4 but 2 + 2 does not equal 5. Thus the factor pair 2, 2 will not work. The factor pair 1, 4 will work since (1)(4) = 4 and 1 + 4 = 5. Thus, x2 + 5x + 4 = (x + 1)(x + 4).

Factor the expression x² - 6x + 8.             x² - 6x + 8 = (x + ?)(x + ?) In place of the question marks, try various factor pairs so that the product of two factors = 8 and the sum of the factors = -6. The factor pairs must be negative in this case; we can consider -1, -8 and -2, -4. (-1)(-8) = 8 but (-1) + (-8) does not equal -6. Thus, the factor pair -1, -8 will not work. The factor pair -2, -4 will work since (-2)(-4) = 8 and (-2) +(- 4) = -6. Thus, x² - 6x + 8 = (x -2)(x - 4).

Another example: Factor the expression 3x² - 9x -54. Check for a greatest common factor; there is one: 3. First we factor out the 3: 3x² - 9x -54 = 3(x² - 3x – 18) = 3(x - ?)(x - ?) What are the factors of -18 that will give us a product of -18 and a middle term of -3? There are several to consider: 1 and -18, -1 and 18, 2 and -9, -2 and 9, 3 and -6, -3 and 6. Which one works? 3 and -6 will work. Thus 3x² - 9x -54 = 3(x² - 3x – 18) = 3(x + 3)(x -6).

It is a little bit more complicated when the coefficient of the x2 term is not equal to 1 (after the greatest common factor has been factored out). In this case, we find factors that meet the following two criteria:

1. The product of the factors = the last term
2. To find the coefficient of the x term, we multiple one of the factors by the coefficient of the x2 term, and add that product to the other multiple.

Factor the expression 2x² + 7x + 6.    2x² + 7x + 6 = (2x + ?)(x + ?) For a product of 6 for the last term, we have (1)(6), (6)(1), (2)(3), or (3)(2). The difficulty is getting 7 for the coefficient of the x term. To get 7 for the coefficient of the x term, which multiple when multiplied by 2, plus the other multiple, equals 7? In this case, (2)(2) + 3 = 7. Thus 2x² + 7x + 6 = (2x + 3)(x + 2).

Remember the following three important relationships:
a² + 2ab + b² = (a+b)(a+b) or (a+b)²
a² -2ab +b² = (a-b)(a-b) or (a-b)²
a² – b² = (a+b)(a-b)

You must memorize these: a² + 2ab + b² , a² -2ab +b² , and a² – b² . They appear again and again on the SAT.

a² + 2ab + b² = what?      (a+b)(a+b) or (a+b)²
a² -2ab +b² = what?        (a-b)(a-b) or (a-b)²
a² – b² = what?                (a+b)(a-b)

Factor the expression     x⁴ – y⁴ .                      x⁴ – y⁴ = (x² + y²)(x² – y²) =  (x² + y²)(x + y)(x - y)

Which of the following is not a factor of x⁵ – x³?                             

1. x³
2. x - 1
3. x² - 1
4. x³ - 1

x⁵ – x³ = x⁵ – x³(x + 1)(x - 1)

Therefore, x³ is a factor, (x-1) is a factor, and (x² – 1) is a factor; (x³ – 1) is not a factor.

Keeping in mind the information above, ansswer the following questions.

1. Factor the following expression: x² – 5x + 6.
   
   
2. (7x + 10) – (4x – 8) is equivalent to
   
   1. 3(x + 6)
   2. 3x + 2
   3. 3(x + 2)
   4. 3(x + 18)

   (7x + 10) – (4x – 8) = 7x + 10 – 4x + 8 = 3x + 18 = 3(x + 6) -------------- A

3. Factor the expression 2x² – 20x – 22
   
   
4. Factor the expression: 3x² + 2x - 5
   
   
5. Where             (x + 3)         is defined, it is equivalent to which of the following expressions?
                  ----------------
                     (x² - 2x - 15)
   
   
   1. 3 / (x - 15)
   2. 1 / (x - 5)
   3. 3 / (x + 5)
   4. 1 / (x + 5)
      
      
6. If (a + b)² = 169 and (a - b)² = 25, then ab = ?

1. Factor the following expression: x² – 5x + 6.
   
   x² – 5x + 6 = (x + ?)(x + ?) 
   In place of the question marks, try various factor pairs so that the product of two factors = 6 and the sum of the factors = -5. The factor pairs must be negative in this case; we can consider -1, -6 and -2, -3. (-1)(-6) = 6 but (-1) + (-6) does not equal -5. Thus, the factor pair -1, -6 will not work. The factor pair -2, -3 will work since (-2)(-3) = 6 and (-2) +(- 3) = -5. Thus, x² - 5x + 6 = (x -2)(x - 3).
   
   
2. (7x + 10) – (4x – 8) is equivalent to
   
   1. 3(x + 6)
   2. 3x + 2
   3. 3(x + 2)
   4. 3(x + 18)

   (7x + 10) – (4x – 8) = 7x + 10 – 4x + 8 = 3x + 18 = 3(x + 6) -------------- A

3. Factor the expression 2x² – 20x – 22
   Check for a greatest common factor; there is one; it is 2. Factor out the 2:
   2x² – 20x – 22 = 2(x² – 10x – 11) = 2(x + ?)(x + ?) Again, in place of the question marks, try various factor pairs so that the product of two factors = -11 and the sum of the factors = -10.
   The factor pairs are (1)(-11) and (-1)(11). In this case (1)(-11) works. Thus,
   2x² – 20x – 22 = 2(x² – 10x – 11) = 2(x + 1)(x -11).
   
   
4. Factor the expression: 3x² + 2x - 5
   
   There is no common factor; thus 3x² + 2x – 5 = (3x + ?)(x + ?) For product of -5 for the last term, we have (1)(-5) or (-1)(5). The difficulty is getting 2 for the coefficient of the x term. To get 2 for the coefficient of the x term, which of the two multiples, when multiplied by 3, plus the other multiple, equals 2? In this case, (3)(-1) + 5 = 2.
   Thus 3x² + 2x -5 = (3x + 5)(x -1).
   
   
5. Where             (x + 3)         is defined, it is equivalent to which of the following expressions?
                  ----------------
                     (x² - 2x - 15)
   
   
   1. 3 / (x - 15)
   2. 1 / (x - 5)
   3. 3 / (x + 5)
   4. 1 / (x + 5)

   To solve this problem, we must factor the denominator. (x² – 2x -15) = (x + ?)(x + ?) What are the factors of -15 that will give us a product of -15 and a middle term of -2? There are several to consider: 1 and -15, -1 and 15; and 3 and -5, -3 and 5. Which one works? 3 and -5 will work since (3)(-5) = -15 and 3 +(-5) = -2. Thus x² - 2x -15 = (x + 3)(x -5).
   
          x + 3               =               =            x + 3                =                         1
   -------------                                 -------------                         -------------
     x² - 2x -15                                  (x + 3) (x - 5)                                  (x - 5)
   
   The answer is B.

6. If (a + b)² = 169 and (a – b)² = 25, then ab = ?

   We start by factoring (a + b)² and (a-b)².
   
   (a + b)² = a² + 2ab + b² = 169
   (a - b)² = a² - 2ab + b² = 25
   
   To solve for ab, multiply both sides of the second equation by -1, and then add the two equations.
   
    a² + 2ab + b² = 169
   -a² + 2ab - b² = -25
   --------------------
            4ab        = 144
              ab        = 36
   
   Note that we did not solve for a and b separately, we solved for ab.

SAT QUICK CHALLENGE EXERCISE R21
The Apostrophe and The Pronoun

The Possessive Pronoun. You will recall that we add an apostrophe and the letter “s” to a noun to show what that noun owns. For example, we write the boy’s wagon or the girl’s bike. However, we do not use apostrophes to show ownership for pronouns. Instead, we use possessive pronouns. Hence, we would say “his” wagon or “her” bike. The “Possessive Pronouns” chart below illustrates how to show ownership for pronouns.
.
The Pronoun Contraction. Although we never use the apostrophe to show pronoun ownership, we must use the apostrophe to make pronoun contractions. For instance, for I am, we write I’m, or for “You are,” we write “You’re.” Remembering how to use the apostrophe correctly for pronouns will enable you to earn points each time an SAT question requires you to show that you have mastered that skill. The “Pronoun Contractions” chart below shows how to write pronoun contractions correctly. After studying the charts, complete Exercise R21 (beneath the chart). Then, use the Answer Key to check your work.

1. D
2. B
3. D

Quadratic Equations: An Introduction

We have looked at linear equations and systems of linear equations. We will next study a third type of equation, quadratic equations. These are equations that have an x2 term, such as x2 + 3x + 2 = 0. There are three ways that quadratic equations can be solved: by factoring, by using the quadratic equation, and by completing the square.

We will review the factoring process. A factor of a number is defined as a positive integer that can be divided evenly unto the number – without a remainder. For example, the factors of 18 are 1,2,3,6,9, and 18.

On the SAT you will have problems dealing with common factors and the greatest common factor. A common factor is a factor that appears in two or more terms. The greatest common factor is the largest of the common factors.

The common factors in 10 and 30 are 1, 2, 5, and 10.

The greatest common factor in 12 and 30 is 6; the common factors are 1, 2, 3, and 6, and the greatest of these is 6.

What is the greatest common factor in 24x³ – 16x² + 8x? It is 8x; 8 is the greatest common factor in 24, 16, and 8, and x is a factor that appears in all three terms.

Factoring a polynomial is the reverse of multiplying (or distributing).

For example, find the product: (3x + 1)(x + 2)

To multiply two binomials, we find the sum of the products by using the F O I L method (First terms, Outer terms, Inner terms, Last terms)

          F              multiply the First terms (3x)(x) = 3x2
          O             multiply the Outer terms (3x)(2) = 6x
          I              multiply the Inner terms (1)(x) = x
          L             multiply the Last terms (1)(2) = 2

Now we add the terms: 3x² + 6x + x + 2 = 3x² + 7x + 2

Here is another example: Find the product of (3a – b)(2a + 4b).
(3a – b)(2a + 4b) = (3a)(2a) + (3a)(4b) + (-b)(2a) + (-b)(4b) = 6a² + 12ab – 2ab – 4b² = 6a² + 10ab – 4b²
                                     F              O               I                L

Review the following three important relationships:

1. (a+b)(a+b) or (a+b)² = a² + 2ab + b²
   the sum of two variables squared = first variable squared plus 2 times the product of the two
                                                               variables plus the second variable squared

2. (a-b)(a-b) or (a-b)² = a² -2ab +b²
   The difference between two variables squared = first variable squared minus 2 times the product of
   the two variables plus the second variable squared

3. (a+b)(a-b) = a² – b²                           the difference between two variables times the sum of the two                                                                     variables = the difference between the squares of the two variables                                                             (sometimes stated as the difference between two squares)
         

You must memorize these: (a+b)² , (a-b)² , (a+b)(a-b). They appear again and again on the SAT.

(a+b)² = what?               a² + 2ab + b²
(a-b)² = what?               a² - 2ab + b²
(a+b)(a-b) = what?        a² – b²

Keeping in mind the information above, answer the following questions.

1. What is the value of -7x(2x³ – 5x + 3)?                
   
   
2. What is the value of (x-4)(3x + 1)?
    
3. What is the value of (5a – 7b²)(5a + 7b²)?
   
   
4. What is the value of (5x – 3)²?
   
   
5. What is the value of 3a(2a – b)²?
   
   
6. What is the value of (3x + y)(x² – y)?

1. -7x(2x³ – 5x + 3) = ?                           Here we distribute
   -7x(2x³ – 5x + 3) = -14x⁴ + 35 x² – 21x
   
   
2. (x-4)(3x + 1) = ?                                    Use F O I L   
   (x-4)(3x + 1) = (x)(3x) + (x)(1) + (-4)(3x) + (-4)(1) = 3x² + x - 12x - 4 = 3x² - 11x – 4
                               F           O             I               L         
    
3. (5a – 7b²)(5a + 7b²) = ?                     Difference of two squares
   (5a – 7b²)(5a + 7b²) = 25a² – 49b⁴
   
   
4. (5x – 3)² = ?                                          The difference between two variables squared
   (5x – 3)² = 25x² – 30x + 9
   
   
5. 3a(2a – b)² = ?                                      The difference between two variables squared; then distribute
   3a(2a – b)² = 3a(4a² – 4ab + b²) = 12a³ – 12a²b + 3ab²
   
   
6. (3x + y)(x² – y) = ?                                  Use F O I L
   (3x + y)(x² – y) = 3x³ – 3xy + x²y – y²
                                F        O        I      L

SAT QUICK CHALLENGE EXERCISE Q21
The Dangling Modifier

The Introductory Modifying Phrase (IMP) and the Subject of a Sentence. You can think of a sentence with an introductory modifying phrase (IMP) as a two-part sentence. The first part is the IMP itself, which modifies the subject of the sentence. The second part is the main clause, which (1) follows the comma after the IMP, (2) begins with the subject, and (3) expresses the main idea. The comma after the IMP connects the IMP to the subject. Note that in Sentence A, which follows, all the sentence elements are where they should be, and the sentence is written correctly. Sentence A. Having forgotten her homework, Marla was afraid that her teacher would think that she had not done it.

Identifying a Dangling Modifier. When the subject an IMP should modify is not included in the sentence, the IMP is sometimes described as “just hanging there” because it is not connected to that subject. For this reason, an IMP in a sentence which does not have the subject the IMP should modify is referred to as a dangling modifier. Note Sentence B, which follows. Sentence B. When driving along Highway 37, smoke from the forest fire nearby made breathing very difficult. Explanation B. As written, the IMP modifies the word “smoke” because “smoke” is the subject of the clause that follows the IMP. The subject that should be in the sentence (a noun that could drive along the highway) is not in the sentence at all, so the IMP is a dangling modifier. You can correct this error by rewriting either the IMP or the main clause, as explained below.

Correcting a Dangling Modifier: Revising Either the IMP or the Main Clause. One way to correct Sentence B’s dangling modifier is (1) to leave the IMP as it is, (2) place the missing subject right after the comma that follows the IMP, and (3) use that subject to express the main idea of the sentence, as in Sentence C, which follows. Sentence C. When driving along Highway 37, Mrs. Cooper found that smoke from the forest fire nearby made breathing very difficult.

Another way to correct the dangling modifier is (1) to add the subject to the IMP, and (2) leave the main clause as it is, as in Sentence D, which follows. Sentence D. When Mrs. Cooper was driving along Highway 37, smoke from the forest fire nearby made breathing very difficult.

Now, keeping in mind the information above, complete the exercise below.

1. B
2. D
3. C

Systems of Linear Equations, Part 6: Word Problems  

In recent years word problems have become more numerous on the SAT. Some word problems will require that you create a system of equations from the information given. Sometimes you will be asked to solve the system of equations, but often you will just be asked to develop the two equations. It is essential, then, that you be able to find and translate the information in the problem into the equations. The following procedure is useful in determining the required information.

Step 1: Read the problem carefully and identify these important items for each equation:

1. The variables
2. The coefficients of the variables
3.  The total

Remember that each of the two equations is in the format: Ax +By = C where x and y are variables,
A and B are coefficients, and C is the total.

The coefficients can be negative or positive;
the coefficients can equal 1;
each equation has a total on the right side of the equation;
and the total can be negative or positive

Step 2. Write the appropriate coefficients and total for each equation.

Step 3. Review the problem to ensure that your equations are consistent with the information in the problem.

Let’s look at the following example:

Albert, Brad, and Charles are selling hamburgers and hot dogs. The hamburgers are $2.50 each and the hot dogs are $1.75 each. On yesterday they had 58 customers each buying only one item and they made $128.50. Which of the following equations will let them know how many hamburgers and how many hot dogs they sold?

1. x + y = 58
   1.75x + 2.50y = 128.50
   
   
2. x + y = 58
   250x + 175y = 128.50
   
   
3. x + y = 58
   2.50x + 1.75y = 128.50
   
   
4. x + y = 128.5
   2.50x + 1.75y = 58

Let's use the steps outlined above:

Step 1: Read the problem carefully and identify these important items for each equation:

1. The variables:  The variables are the number of hamburgers and the number of hot dogs.  Let x = the number of hamburgers and y = the number of hot dogs1. The variables:  The variables are the number of hamburgers and the number of hot dogs.  Let x = the number of hamburgers and y = the number of hot dogs
2. The coefficients of the variables:  in one equation the coefficient of x is 1 and the coefficient of y is 1; in the other equation the coefficient of x is 2.50 and the coefficient of y is 1.75.
3. The total:  the total for one equation is 58 the total for the other equation is 128.50

Step 2. Write the appropriate coefficients and total for each equation.
x - y = 1
1
x + y = 59

Step 3. Review the problem to ensure that your equations are consistent with the information in the problem.

The equations are consistent.

Keeping in mind the information above, answer the following system of equations questions.

1. Delta State University has separate tuition rates for in-state students and out-of-state students.  In-state students are charged $421 per semester and out-of-state students are charged $879 per semester.  The university’s junior class of 1,980 students paid a total of $1,170,210 in tuition fees for the most recent semester. Which of the following systems of equations represents the number of in-state (x) and out-of-state  (y) juniors and the amount of tuition fees the two groups paid?
   1. x + y  =  1,170,210
      421x + 879y  =  1,980
      
      
   2. x + y  =  1,980
      879x + 421y  =  1,170,210
      
      
   3. x + y  =  1,980
      421x + 879y  =  1,170,210
      
      
   4. x + y  =  1,170,210
      879x + 421y  =  1,980
      
      
2. An online bookstore sells novels and magazines.  Each novel sells for $4, and each magazine sells for $1.  If Jackie purchased a total of 11 novels and magazines that have a combined selling price of $20, how many novels did she purchase?
   1. 2
   2. 3
   3. 4
   4. 5
      
      
3. A bead shop sells wooden beads for $0.20 each and crystal beads for $0.50 each.   If a jewelry artist buys 127 beads total and pays $41 for them, how much more did she spend on crystal beads than wooden beads?
   
   1. $11
   2. $15
   3. $23
   4. $26

1. Delta State University has separate tuition rates for in-state students and out-of-state students.  In-state students are charged $421 per semester and out-of-state students are charged $879 per semester.  The university’s junior class of 1,980 students paid a total of $1,170,210 in tuition fees for the most recent semester. Which of the following systems of equations represents the number of in-state (x) and out-of-state  (y) juniors and the amount of tuition fees the two groups paid?
   
   
   1.  x + y  =  1,170,210
      421x + 879y  =  1,980
      
      
   2.  x + y  =  1,980
      879x + 421y  =  1,170,210
      
      
   3. x + y  =  1,980
      421x + 879y  =  1,170,210
      
      
   4. x + y  =  1,170,210
      879x + 421y  =  1,980

   In this case we are not asked to solve the system of equations; just determine what they are. Use the process described earlier:
   
   
   Step 1: Read the problem carefully and identify these important items for each equation:
   
   - The variables:  The variables are the number of in-state juniors and the number of
      out- of-state juniors.  Let x = the number of in-state juniors and y = the number of
      out-of-state juniors. 
   - The coefficients of the variables:  in one equation the coefficient of x is 1 and the
      coefficient of y is 1; in the other equation the coefficient of x is 421 and the
      coefficient of y is 879.
   - The total:  the total for one equation is 1,980 and the total for the other equation is
      1,170,210.
   
   Step 2. Write the appropriate coefficients and total for each equation.
   
     x + y = 1.980
     421x + 879y = 1,170,210             The answer is C.

2. An online bookstore sells novels and magazines.  Each novel sells for $4, and each magazine sells for $1.  If Jackie purchased a total of 11 novels and magazines that have a combined selling price of $20, how many novels did she purchase?

   1. 2
   2. 3
   3. 4
   4. 5

   In this case we are asked to determine the system of equations and then solve the problem. Use the process described earlier to determine the system of equations,
   
   Step 1:  Read the problem carefully and identify these important items for each equation:
   -  The variables:  The variables are the number of novels and the number of         
       magazines.  Let x = the number of novels  and  y = the number of magazines. 
   -  The coefficients of the variables:  in one equation the coefficient of x is $4 and the
       coefficient of y is $1; in the other equation the coefficient of x is 1 and the
       coefficient of y is 1.
     - The total:  the total for one equation is $20 and the total for the other equation is
        11.
   
   Step 2. Write the appropriate coefficients and total for each equation.
   4x + y = 20
   x + y = 11
   
   Now we can solve the system of equations; let’s use the elimination approach. Multiply the second equation by -1: -x - y = -11. Then
   4x + y = 20
   -x - y = -11
   3x      = 9
          x = 3                The answer is B.

3. A bead shop sells wooden beads for $0.20 each and crystal beads for $0.50 each.   If a jewelry artist buys 127 beads total and pays $41 for them, how much more did she spend on crystal beads than wooden beads?           

   1. $11
   2. $15
   3. $23
   4. $26

   In this case we are asked to determine the system of equations and then solve the problem. Use the process described earlier to determine the system of equations,

   Read the problem carefully and identify these important items for each equation:

   1. The variables:  The variables are the number of wooden beads and the number of crystal beads.     Let x = the number of  wooden beads  and  y = the number of crystal beads.
   2. The coefficients of the variables:  in one equation the coefficient of x is $0.20 and the coefficient of y is $0.50; in the other equation the coefficient of x is 1 and the coefficient of y is 1.2. The coefficients of the variables:  in one equation the coefficient of x is $0.20 and the coefficient of y is $0.50; in the other equation the coefficient of x is 1 and the coefficient of y is 1.
   3. The total:  the total for one equation is $41 and the total for the other equation is 127.
   
   Step 2. Write the appropriate coefficients and total for each equation.
   
   0.2x + 0.5 y = 41
   x + y = 127
   
   Now we can solve the system of equations; let’s use the elimination approach. Multiply the second equation by -0.5: -0.5x - 0.5 y = -63.5. Then
     0.2x + 0.5y =   41
   -0.5x – 0.5y = -63.5
   --------------------
   -0.3x            = -22.5
                     x = 75               = number of wooden beads
   
   

     x + y = 127
   75 + y = 127
            y = 127 – 75 = 52 = number of crystal beads

   Cost of crystal beads = $0.5 (52)   = $26
   Cost of wooden beads = $0.2 (75) = $15
                                                               ----
                                                               $11      The answer is A

SAT QUICK CHALLENGE EXERCISE P21
Placing Modifiers Correctly

The Introductory Modifying Phrase (IMP). An introductory modifying phrase (IMP) (1) comes at the beginning of a sentence, (2) is followed by a comma, and (3) describes the subject of the sentence, which comes right after the IMP (or as close as possible to it). Note Sentence A, which follows. Sentence A. Feeling exhausted after the marathon, Carla just wanted to lie down. Note the bold, underlined IMP at the beginning of the sentence, the comma that follows the IMP, and the bold, italicized subject right after the comma -- Carla. The IMP modifies that subject.

The Modifier That Comes After the Word It Modifies. Unlike the IMP, some modifiers follow the word they modify. At times, a “follower” modifier comes after a comma. At other times, a “follower” modifier is enclosed in parentheses. Some students think that information in parentheses is unimportant and ignore it, but such students taking the SAT have lost points because of that mistake. Do not let that happen to you! Remember that whether it follows a comma or is enclosed in parentheses, a modifier that is placed as close as possible to the word it modifies is always placed correctly. Note Sentences B and C below.

Sentence B.. Marti says that she wants to be an aesthetician, a specialist trained in skin beautification.
Sentence C. Marti says that she wants to be an aesthetician (a specialist trained in skin beautification).

Comments. Although the modifier in Sentence B follows a comma, and the modifier in Sentence C is in parentheses, (1) each modifier modifies the noun that it follows ("aesthetician"), (2) each modifier is in the right place, and (3) both sentences are written correctly.

Now, keeping in mind the information above, complete the exercise below.

1. D
2. C
3. B

Systems of Linear Equations, Part 5: One Solution No Solution, An Infinite Number of Solutions

Most of the system of linear equations problem on the SAT will have one solution. Occasionally, however, there will be a problem where there is no solution or there is an infinite number of solutions. For test takers it is important to be able to determine which situation exists. A standard approach to make this determination is to express each equation in slope-intercept form: y = mx + b, where “m” is the slope and “b” is the y-intercept. In other words, solve each equation for y.

You will also recognize that “y = mx + b” is the formula for a line, and thus there is a line for each of the two equations in a system of linear equations. The slopes and y-intercepts of the two lines determine whether there is one solution, no solution, or an infinite number of solutions to the system of linear equations, as follows:

One solution.

If the two lines have different slopes, there will be one solution to the system of equations. Geometrically, the two lines will cross, and the point of intersection (x,y) will be the solution to the system of equations.

No solution.

If the two lines have the same slope but different y-intercepts, there will be no solution to the system of equations. Geometrically, the two lines are parallel and will never cross; thus there is no point of intersection, and there is no solution to the system of equations.

An Infinite Number of Solutions.

If the two lines have the same slope and the same y-intercept, there will be an infinite number of solutions to the system of equations. Geometrically, the two lines are really the same line, and one is a multiple of the other. Here is an example:

y = 2x + 7
2y = 4x + 14          The second equation is a multiple of the first one.

The coefficients of the variables and y-intercept of one equation are multiples of the coefficients in the other equation. In this case:
Variable y: 2 in the second equation is a multiple of the 1in the first equation
Variable x: 4 in the second equation is a multiple of the 2 in the first equation
y-intercept: 14 is a multiple of 7             The common multiple is 2.

Keeping in mind the information above, answer the following system of equations questions.

1. 6x + 2y = 3
   3x + y = 2
   How many solutions (x,y) are there to the system of equations above?
   
   
   1. Zero
   2. One
   3. Two
   4. More than two
      
      
2. 8x – 2y = 12
   6y = kx – 42
   In the system of equations above, k represents a constant.  If the system of equations has no solution, what is the value of 2k?
   
   1. 6
   2. 12
   3. 24
   4. 48

3.     3x – 2y = a
   -15x + by = 10

   In the system of equations above, a and b are constants. If the system of equations has infinitely many solutions, what is the value of a?

1. 6x + 2y = 3
   3x + y = 2
   How many solutions (x,y) are there to the system of equations above?
   
   
   1. Zero
   2. One
   3. Two
   4. More than two

   To determine whether there is one solution, no solution, or an infinite number of solutions, express
   each equation in slope-intercept form (solve for y) and compare the slopes and y-intercepts.

   6x + 2y = 3                                           3x +y = 2
   y = -3x + 1.5                                               y = -3x + 2

   The two equations have the same slope (-3) but different y-intercepts (1.5 and 2). Thus there is no solution to the system of equations; the lines are parallel. The answer is A.

   Let’s see what happens if we solve the equations by substitution. Solve the second equation for y:
   y = -3x + 2. Now substitute this value of y in the first equation and solve for x:
   6x + 2(-3x + 2) = 3
   6x – 6x + 4 = 3
   4 = 3                           Of course this is not true; thus there is no solution.

2. 8x – 2y = 12
   6y = kx – 42
   In the system of equations above, k represents a constant.  If the system of equations has no solution, what is the value of 2k?
   
   1. 6
   2. 12
   3. 24
   4. 48
   
   There is no solution if the slopes are equal and the y-intercepts are unequal.  Let’s express each equation in slope-intercept form.There is no solution if the slopes are equal and the y-intercepts are unequal.  Let’s express each equation in slope-intercept form.
   
   8x – 2y = 12     
         -2y = -8x + 12                                             
             y = 4x - 6 
           
      The y-intercepts are unequal; now we need to determine whether the slopes are equal.  They are equal if 4 = k/6. 
   Thus we need to solve for k.
   k/6 = 4                  
      k = 24         If the system of equations has no solution, then k must = 24.   The question asks us for 2k.  Thus the answer is 2(24) = 48  -------- D
   
   

3.     3x – 2y = a
   -15x + by = 10

   In the system of equations above, a and b are constants. If the system of equations has infinitely many solutions, what is the value of a?

   If a system of equations has infinitely many solutions, the two lines have the same slope and the same y-intercept. The two lines are really the same line, and one is a multiple of the other.
   
   There is a common multiple that is applied to each of the two variables (x and y) and to the y-intercept.

   In this case the second equation is a multiple of the first equation. What is the common multiple?

   Look at the coefficients of x, the coefficients of y, and the y-intercepts. Is there a multiple in either of these?
   Yes, there is a multiple in the x coefficients; -15 is a multiple of 3.

   The multiple is -5 since 3(-5) = -15. -5 is the common multiple, and we will use it to determine b and a.

   – 2(-5) = b                                        a(– 5) = 10
   b = 10                                                      a = 10/(– 5) = – 2

   
   We can check our work by calculating the slope and y-intercept for each equation using the values of a and b that we calculated. The slopes and y-intercepts should be the same for both equations.

   First equation: 3x – 2y = a
   3x – 2y = – 2
        – 2y = – 3x – 2
              y = 1.5x + 1
   
   Second equation:             – 15x + by = 10
                                           – 15x + 10y = 10
                                                        10y = 15x + 10
                                                            y = 1.5x + 1

   The slopes and y-intercepts are the same.

SAT QUICK CHALLENGE EXERCISE O21
The Relative Pronoun

Using “When.” Use when to refer to times and events. Note the following: Do you know the dates where homecoming will be celebrated this year? Explanation. That sentence is incorrect. Since it talks about time, we need a word that refers to time – when, not where. Correction. Do you know the dates when homecoming will be celebrated this year?

Using a Preposition (in, during, to, etc.) and “Which” Instead of “When.” Besides using “when,” you can refer to time by using a preposition that relates to time, followed by the word “which.” Note the following: Do you know the dates during which homecoming will be celebrated this year?

Using “When.” Use when to refer to times and events. Note the following: Do you know the dates where homecoming will be celebrated this year? Explanation. That sentence is incorrect. Since it talks about time, we need a word that refers to time – when, not where. Correction. Do you know the dates when homecoming will be celebrated this year?

Using a Preposition (in, during, to, etc.) and “Which” Instead of “When.” Besides using “when,” you can refer to time by using a preposition that relates to time, followed by the word “which.” Note the following: Do you know the dates during which homecoming will be celebrated this year?

Using Where. Use where to refer to places. Do not use it to refer to times, books, films, etc. Note the following: Disneyland is a place when lots of people like to go during the summer. Explanation. Disneyland is a place, and we need a word that refers to places – where, not when. Correction. Disneyland is a place where lots of people like to go during the summer.

Using a Preposition (in, during, to, etc.) and “Which” Instead of “Where.” Besides using the word“ where,” you can refer to places by using a preposition that relates to places, followed by the word “which.” Note the following: New York is a city in which many people travel by subway every day.

. Use where to refer to places. Do not use it to refer to times, books, films, etc. Note the following: Disneyland is a place when lots of people like to go during the summer. Explanation. Disneyland is a place, and we need a word that refers to places – where, not when. Correction. Disneyland is a place where lots of people like to go during the summer.

Using a Preposition (in, during, to, etc.) and “Which” Instead of “Where.” Besides using the word“ where,” you can refer to places by using a preposition that relates to places, followed by the word “which.” Note the following: New York is a city in which many people travel by subway every day.

Now, keeping in mind the information above, complete the exercise below.

1. B
2. B
3. A

Systems of Linear Equations, A Short Cut - Part 4

The traditional method of determining the answer in a system of linear equations problem involves calculating the values of each of the two variables by substitution, by elimination, or by using the answer choices. In some cases it is possible to find the answer without determining the values of each of the two variables. Consider the following example:
3x + 2y = 8
x - 4y = 12

Based on the system of equations above, what is the value of 4x - 2y?
Let’s solve this problem by substitution.

Step 1: Solve for x in the second equation.
x - 4y = 12
x = 12 + 4y

Step 2. Substitute this expression of x in the first equation, and solve for y.
3x + 2y = 8
3(12 + 4y) + 2y = 8
36 + 12y + 2y = 8
14y = 8 - 36
14y = -28
y = -2

Step 3. Substitute this value of y in the second equation, and solve for x.
x - 4y = 12
x - 4(-2) = 12
x + 8 = 12
x = 4

Now that we have values for x and y, we can find the value of 4x – 2y:
4x – 2y = 4(4) – 2(-2) = 16 + 4 = 20

In this case an alternative for finding our answer is simply to add the two equations:

3x + 2y = 8
x - 4y = 12
------------
4x – 2y = 20

We get 4x – 2y = 20, the same answer we got using the longer traditional approach. Whenever you have a problem similar to this one (where you are asked questions such as find the value of 3x + 5y or 8x + 10y or 3x + 7y), try adding the two equations or subtracting one equation from the other one.

Keeping in mind the information above, answer the following systems of equations questions.

1. 2x + 3y = 2
   x – 4y = 12
   Based on the system of equations above, what is the value of  3x – y?

2. 4x + y = 14
   3x + 2y = 13
   Based on the system of equations above, what is the value of x - y?
   
   
3. 2x + y = 6
   7x + 2y = 27
   Based on the system of equations above, what is the value of 3x + y?

1. 2x + 3y = 2
   x – 4y = 12
   Based on the system of equations above, what is the value of  3x – y?
   
   

   Solving this system of equations be substitution or by elimination, we find that x = 4 and y = -2; thus 3x – y = 3(4) – (-2) = 12 + 2 = 14

   The short cut, which will work in this case, is to add the two equations together:

   2x + 3y = 2
   x – 4y = 12
   ------------
   3x –y = 14             By adding, we get 3x – y = 14, which is our answer; we did not need to determine
                                 the individual values of x and y.

2. 4x + y = 14
   3x + 2y = 13
   Based on the system of equations above, what is the value of x - y?
   
   

   Try adding the equations:
   4x + y = 14
   3x + 2y = 13
   -------------
   7x + 3y = 27

   Adding did not work. Try subtracting one equation from the other one. Subtract the second equation from the first equation.

   4x + y = 14
   -(3x + 2y = 13)
   ---------------

   4x + y = 14
   -3x - 2y = -13
   x – y = 1                  This is our answer, and again we did not need to determine the individual values
                                   of x and y. (For your information, if you choose to calculate the values of x
                                   and y, x = 3, y = 2.)

3. 2x + y = 6
   7x + 2y = 27
   Based on the system of equations above, what is the value of 3x + y?
   
   

   Try adding the equations:
   2x + y =     6
   7x + 2y = 27
   ___________
   9x + 3y = 33

   This is not our answer, but we observe that 9x + 3y is a multiple of what we are trying to find, 3x + y.

   We can divide the 9x + 3y = 33 by 3, and we get 3x + y = 11. Thus 11 is our answer.
   (For your information, if you choose to calculate the values of x and y, x = 5, y = -4.)

SAT QUICK CHALLENGE EXERCISE N21
The Relative Pronoun

You will recall that a clause is a group of words that contains a subject and a verb. An independent clause expresses a complete thought and can stand alone as a complete sentence, but a dependent clause is simply a part of a sentence. It does not express a complete thought and cannot stand alone. A dependent clause that begins with a relative pronoun and modifies a noun in the independent clause of a sentence is called a relative clause. The tips below can help you remember how to use selected relative pronouns correctly (which, that, who, and whom). You might see questions about them when you take the SAT.

Using That and Which. The word that can modify both things and people. It introduces essential clauses, so no comma should come before it because essential clauses must not be separated from the rest of the sentence. Note Examples A and B, which follow. Example A. Leon is very excited about the letter, that he received from his dad today. Explanation A. This sentence is incorrect because a comma separates the relative pronoun “that” from the rest of the sentence. Note the correction in Example B, which follows. Example B. Leo is excited about the letter that he received from his dad today.

The word which modifies things only. When which introduces a nonessential clause, that clause must be enclosed in commas, but when it introduces an essential clause, commas must not separate the clause from the rest of the sentence. Note Examples C and D, which follow. Example C. Pecans which are very nutritious are one of my favorite snacks. Explanation C. Example C is incorrect because commas must separate the nonessential clause from the rest of the sentence. Note the correction in Example D. Example D. Pecans, which are very nutritious, are one of my favorite snacks. *CAUTION: Do not use commas for which when that word means which ones, as in the following sentence: Mrs. Lee said to give some of these canned goods to the soup kitchen, but she didn’t say which (ones) to give away. In this situation, the relative clause is quite essential and must not be separated from the rest of the sentence.

Using Who and Whom. Who and whom both refer to people. Who must be used before a verb, and whom must be used after a preposition (such as about, for, from, on, to, or with). Note Examples E and F, which follow. Example E. Calvin Wilkins, whom coached at Rynolds College for 12 years, is our new coach. Explanation E. Example E is incorrect because who must be used before a verb. Note the correction in Example F, which follows. Example F. Calvin Wilkins, who coached at Reynolds College for 12 years, is our new coach. Finally, note Example G, which follows. Example G. His wife Helena is the person for who Marvin wrote this poem. Explanation G. The word who is incorrect. We need a relative pronoun that (1) refers to people and (2) follows a preposition. Note that Example H fulfills both requirements by using the word whom. Example H. Marvin’s wife Helena is the person for whom he wrote this poem.
Now, keeping in mind the information above, complete the exercise below.

1. C
2. D
3. B

System of Linear Equations - Part 3

A system of equations can be solved in three ways: by substitution, by elimination, and by using the answer choices. When elimination is used, we find a way to eliminate one of the variables from the calculation. The first step is to multiply one of the equations by a number so that when the two equations are added together, one of the variables falls out. (For example, we might get 6x – 6x = 0 or 8y – 8y = 0.) We then solve for the remaining variable, then use the value of that variable to find the value of the variable that was eliminated.

Consider the following example:

3x + 4y = -23
-x + 2y = -19

Which ordered pair (x,y) satisfies the system of equations above?

1. (-5, -2)
2. (3, -8)
3. (4, -6)
4. (9, -6)

Step 1. To eliminate one of the variables, we could multiply the second equation by 3 so that the x variable would be eliminated (we would get 3x – 3x = 0). Or we could multiply the second equation by -2 so that the y variable could be eliminated (we would get 4y – 4y = 0).

In this case, multiply the second equation by 3.

-x + 2y = -19 -----------  -3x + 6y = -57

Step 2. Add the two equations together and solve for y.

 3x + 4y = -23
-3x + 6y = -57
---------------
          10y = -80 (The x variable is eliminated.)
               y = -8

Step 3. Substitute this value of y into either of the two equations and solve for x.
First equation: 3x + 4y = -23
3x + 4(-8) = -23
3x - 32 = -23
3x = 9
x = 3

Second equation: -x + 2y = -19
-x + 2(-8) = -19
-x - 16 = -19
-x = -3
x = 3

Thus our answer choice is B: x = 3 and y = -8.

Keeping in mind the information above, solve the following systems of equations by elimination.

1.   x + y  =  0                    
   3x - 2y = 10
   Which of the following ordered pairs (x,y) satisfies the system of equations above?

   1. (3, -2)
   2. (2, -2)
   3. (-2, 2)
   4. (-2, -2)

2.   x + 2y = 7
   2x + 3y = 11
   Which of the following ordered pairs (x,y) satisfies the system of equations above?

   1. (-1, -3)
   2. (-1, 3)
   3. (1, 3)
   4. (1, -3)

3.  -2x = 4y + 6
   2(2y + 3) = 3x - 5
   Which of the following ordered pairs (x,y) satisfies the system of equations above?

   1. (1, 2)
   2. (1, -2)
   3. (-1, -1)
   4. (-1, 1)

1.   x + y  =  0                    
   3x - 2y = 10
   Which of the following ordered pairs (x,y) satisfies the system of equations above?

   1. (3, -2)
   2. (2, -2)
   3. (-2, 2)
   4. (-2, -2)
      
      

      Step 1. Multiply the first equation by 2 so that the y variable can be eliminated.
      x + y = 0 ----------- 2x + 2y = 0
      
      Step 2. Add the two equations together and solve for x.
      2x + 2y = 0
      3x - 2y = 10
      ------------
      5x          = 10
                  x = 2

      Step 3. Substitute this value of x into either of the two equations and solve for y.
      First equation:
      x + y = 0
      2 + y = 0
      y = -2
      
      Second equation:
      3x - 2y = 10
      3(2) -2y = 10
      6 – 2y = 10
      -2y = 4
      y = -2
      
      Thus our answer choice is B: x = 2 and y = -2.

      
      
      

2.   x + 2y = 7
   2x + 3y = 11
   Which of the following ordered pairs (x,y) satisfies the system of equations above?

   1. (-1, -3)
   2. (-1, 3)
   3. (1, 3)
   4. (1, -3)
      
      

      Step 1. Multiply the first equation by -2 so that the x variable can be eliminated.
      x + 2y = 7 -----------     -2x - 4y = -14                        
      
      Step 2. Add the two equations together and solve for x.
      -2x - 4y = -14
      2x + 3y = 11
      ------------------
      -y = -3
      y = 3

      Step 3. Substitute this value of y into either of the two equations and solve for x.
      First equation: x + 2y = 7
      x +2(3) = 7
      x + 6 = 7
      x = 1
      
      Second equation: 2x +3y = 11
      2x + 3(3) = 11
      2x+ 9 = 11
      2x = 2
      x = 1
      
      Thus our answer choice is C: x = 1 and y = 3.

      
      
      

3.  -2x = 4y + 6
   2(2y + 3) = 3x - 5
   Which of the following ordered pairs (x,y) satisfies the system of equations above?

   1. (1, 2)
   2. (1, -2)
   3. (-1, -1)
   4. (-1, 1)
      

      We first should rewrite both equations in standard form: ax + by = c
      -2x = 4y + 6 2(2y + 3) = 3x - 5
      -2x -4y = 6 4y + 6 = 3x – 5
      -3x + 4y = -11
      Now we have this system of equations:
      -2x -4y = 6
      -3x + 4y = -11
      
      Step 1. In this case we do not need to multiply either equation by a number since the two equations can be added and the y variable can be eliminated.
      
      Step 2. Add the two equations together and solve for x.
      -2x - 4y = 6
      -3x + 4y = -11
      ---------------
      -5x         = -5
                  x = 1

      Step 3. Substitute this value of x into either of the two equations and solve for y.
      First equation: -2x -4y = 6
      -2(1) - 4y = 6
      -2 - 4y = 6
      - 4y = 8
      y = -2
      
      Second equation: -3x +4y = -11
      -3(1) + 4y = -11
      -3 + 4y = -11
      4y = -8
      y = -2
      
      Thus our answer choice is B: x = 1 and y = -2.

SAT QUICK CHALLENGE EXERCISE M21
Using the Dash Correctly

Using the Dash To Set Off Non-Essential Information. You may recall that like the colon, the dash can be used to indicate that a list or an explanation will follow. Similarly, as in the case of the comma, the dash can be used to separate a non-essential phrase or clause from the rest of a sentence. (Re-minder: Non-essential information provides additional insight, but is not needed for expressing the main idea.) Note Example A, which follows. Example A. Jerry’s chorus teacher -- who has a powerful sing-ing voice -- will be a soloist at the State Symphony Orchestra’s annual “Spring Singsation.” Note that since the non-essential information is entirely within the sentence, rather than at the beginning or end, you need two dashes to separate that information from the rest of the sentence, just as you would use two commas if you had chosen to use commas in this situation. However, remember that you must not mix and match commas and dashes by using one of each.

Using the Dash To Create a Deliberate Pause. Another function of the dash is to indicate a deliberate pause in order to highlight or emphasize a strong point. The pause may sometimes lead to a sense of drama, suspense, or humor. Note Example B, which follows. Example B. Baby Lori likes most vegetables, but not spinach – not one bit!  

Use only one pair of dashes per sentence if the non-essential addition is completely within the sentence, but a single dash otherwise. Too many dashes can cause your message to become rather choppy, and understanding it could become quite challenging, as in Example C, which follows. Example C. Aunt Alisa – my favorite aunt of all – perhaps except for Aunt Loretta – or maybe Aunt Cindy – will have a birthday next week, and I plan to give her a surprise party on Friday – because she has never had one – and I think she deserves one. Note the improvement in Example D, which follows. Example D. We’d like for you to come to Aunt Alisa’s birthday party next Friday – her very first surprise party, so shhh!

Now, keeping in mind the information above, complete the exercise below.

1. C
2. B
3. D

Systems of Linear Equations - Part 2

As noted last week, a system of equations can be solved in three ways: by substitution, by elimination, and by using the answer choices. The alternative of using the answer choices can be used only when the answer choices include  values for both variables. Remember answer choices are not given for 5 of the problems in Section 3 of the SAT and answer choices are not given for 8 of the problems in Section 4. Also in some Problems the question might be “What is the value of x?” or “What is the value of x + y?” In these cases we would have to solve the problem by substitution or by elimination.
Let’s look at the substitution process.

Consider the following example:
3x + 4y = -23
-x + 2y = -19

Which ordered pair (x,y) satisfies the system of equations above?

1. (-5, -2)
2. (3, -8)
3. (4, -6)
4. (9, -6)

These are the steps for solving by substitution.

Step 1: In one of the equations solve for one of the variables in terms of the other variable. Let’s solve for x in the second equation.
3x + 4y = -23
         -x = -19 - 2y
           x = 19 + 2y

Step 2. Substitute this expression of x in the first equation, and solve for y.
3x + 4y = -23
3(19 + 2y) + 4y = -23
    57 + 6y + 4y = -23
                   10y = -23 - 57
                   10y = -80
                       y = -8

Step 3. Substitute this value of y in the second equation, and solve for x.
-x + 2y = -19
-x + 2(-8) = -19
-x -16 = -19
        -x = -3
          x = 3                                                  Thus the answer choice is B, x = 3 and y = -8.

Keeping in mind the information above, solve the following systems of equations by substitution.

1. x + y = 0
   3x - 2y = 10
   
   Which of the following ordered pairs (x,y) satisfies the system of equations above?
   
   1. (3, -2)
   2. (2, -2)
   3. (-2, 2)
   4. (-2, -2)
      
      
2.  x + 2y  =  7 
   2x + 3y =  11
   
   Based on the system of equations above, what is the value of x?  (No answer choices are given here.)                          

3. -2x = 4y + 6                                               
   2(2y + 3) = 3x - 5
   
   If the ordered pair (x, y) that satisfies the system of equations is (x, y) what is the value of (x + y)?

   1. -1
   2. 0
   3. 1
   4. 2

1. x + y = 0
   3x - 2y = 10
   Which of the following ordered pairs (x,y) satisfies the system of equations above?
   
   1. (3, -2)
   2. (2, -2)
   3. (-2, 2)
   4. (-2, -2)
   
   These are the steps for solving by substitution.
   
   Step 1: In one of the equations solve for one of the variables in terms of the other variable. Let’s solve for x in the first equation.
   x = -y
   
   Step 2. Substitute this expression of x in the second equation, and solve for y.
   3x - 2y = 10
   3(-y) - 2y = 10
   -3y -2y = 10
          -5y = 10
              y = -2
   

   Step 3. Substitute this value of y in the first equation, and solve for x.
   x + y = 0
    x -2 = 0
          x = 2

   Thus our answer choice is B, x = 2 and y = -2.

   
   
   
2.  x + 2y  =  7 
   2x + 3y =  11
   
   Based on the system of equations above, what is the value of x?  (No answer choices are given here.)
   
   These are the steps for solving by substitution.
   
   Step 1:  In one of the equations solve for one of the variables in terms of the other variable.  Let’s solve for x in the first equation.
                 x = -2y + 7 
   
   Step 2. Substitute this expression of x in the second equation, and solve for y.
   2x + 3y = 11
   2(-2y + 7) + 3y = 11
   -4y + 3y = 11 -14
             -y = -3
               y = 3
   

   Step 3. Substitute this value of y in the first equation, and solve for x.

     x + 2y = 7
   x + 2(3) = 7
       x + 6 = 7
             x = 1                                                                     Thus, our answer is: x = 1

   1. -1
   2. 0
   3. 1
   4. 2

3. We first should rewrite both equations in standard form: ax + by = c
   -2x = 4y + 6                                                2(2y + 3) = 3x - 5
   -2x -4y = 6                                                 4y + 6 = 3x – 5
                                                                   -3x + 4y = -11
   Now we have this system of equations:
    -2x -4y = 6
   -3x + 4y = -11

   These are the steps for solving by substitution.
   
   Step 1: In one of the equations solve for one of the variables in terms of the other variable. Let’s solve for x in the first equation.
   -2x -4y = 6
          -2x = 4y + 6
              x = -2y -3

   Step 2. Substitute this expression of x in the second equation, and solve for y.
   -3x + 4y = 6
   -3(-2y -3) + 4y = -11
           6y +9 + 4y = -11
                      10y = -20
                           y = -2

   Step 3. Substitute this value of y in the first equation, and solve for x.
   -2x -4y = 6
   -2x -4(-2) = 6
   -2x +8 = 6
        -2x = -2
            x = 1
   
   Now we have values for x and y: x = 1 and y = -2.
   Thus x + y = 1 + (-2) = -1, and our answer choice is A.

SAT QUICK CHALLENGE EXERCISE L21
The Colon and the Dash

The Colon. One role of the colon is to introduce a list, but what comes before the colon must be a complete sentence. What comes after the colon can be either a sentence or a fragment. Do not capitalize the first word after a colon unless there is a grammatical reason to capitalize it (if it is a proper noun or the first word of a complete sentence).

Avoid answers with “including” or “such as” before a colon; such wording will not be a complete sentence. Note examples A1 and A2, which follow. Example A1. Ella makes yummy desserts, including: cakes, pies, cookies, and even candy. (Incorrect) Example A2. Ella makes yummy desserts: cakes, pies, cookies, and even candy. (Correct)

The colon can also be used just before an explanation, and a complete sentence typically follows that colon. Since a colon used this way does the same job as a semicolon, a period, or a dash, the SAT generally will not ask you to select from among those identical choices. However, if you are faced with that situation, the colon will be correct only if the sentence following the colon explains what was stated before the colon. Note Examples B1 and B2, which follow. Example B1: We will watch the NCAA championship game on my friend’s big-screen TV: that game is always exciting. (Incorrect because the sentence after the colon does not explain the decision about watching the game. Example B2: We will watch the NCAA championship game on my friend’s big-screen TV: we’ll save lots of time and money, and we’ll still have a great time. (Correct because the sentence after the colon does explain the decision to watch the game on the friend’s big-screen TV.)

The Dash. Like the colon, the dash can be used before a list or an explanation. Examples C1 and C2, below, show how to use the dash to write correctly the same sentences written correctly with colons in Examples A2 and B2 above. Example C1. Ella made some delicious desserts -- cakes, pies, cookies, and candy. Example C2. We will watch the NCAA championship game on my friend’s big-screen TV – We’ll save lots of time and money, and we’ll still have a great time.

Now, keeping in mind the information above, complete the exercise below.

1. C
2. B
3. D

Systems of Linear Equations - Part 1

One of the types of questions you will see on the SAT is solving systems of linear equations, sometimes called simultaneous equations or two equations in two unknowns.  A system of equations can be solved in three ways: by substitution, by elimination, and by using the answer choices (when answer choices are given).  When the alternative of using the answer choices is used, we substitute an answer choice in each of the two equations and determine which one works.  The correct answer choice must work in both equations.  Consider the following example:

3x + 4y = -23
-x + 2y  =  -19

Which ordered pair (x,y) satisfies the system of equations above?

1. (-5, -2)
2. (3, -8)
3. (4, -6)
4. (9, -6)  

Try choice A):  x = -5 and y = -2
First equation:   
3x +  4y  =  -23                                                                                                      3(-5) + 4(-2)  =  -23                           
-15  -8  =  -23           YES            

Second equation:   
-x +  2y  =  -19                                                                                                     
-(-5) + 2(-2)  =  -19                             
5  -4  = 1, not -19         NO; choice A does not work; it needs to work in both equations.

Now try choice B):  x = 3 and y = -8
First equation:   
3x +  4y  =  -23                                                                                                                                                                                                    
3(3) + 4(-8)  =  -23                             
9  -32  =  -23           YES            

Second equation:   
-x +  2y  =  -19                                                                                                                                                                                                      
-(3) + 2(-8)  =  -19                               
-3  -16  =  -19        YES           
Thus our answer is B; these numbers work for both equations. Answer choices C and D do not work..

Keeping in mind the information above, solve the following systems of equations by using the answer choices.

1. x + y = 0
   3x - 2y = 10

   Which of the following ordered pairs (x,y) satisfies the system of equations above?

   1. (3, -2)
   2. (2, -2)
   3. (-2, 2)
   4. (-2, -2)
   
   
   
2. x + 2y  =  7
   2x + 3y =  11
   Which of the following ordered pairs (x,y) satisfies the system of equations above?
   
   1. (-1, -3)
   2. (-1, 3)
   3. (1, 3)
   4. (1, -3)
3. -2x = 4y + 6
   -2x = 4y + 6                           
   2(2y + 3) = 3x - 5
   
   Which of the following ordered pairs (x,y) satisfies the system of equations above?
   
   1. (1, 2)
   2. (1, -2)
   3. (-1, -1)
   4. (-1, 1)

1. x + y = 0
   3x - 2y = 10

   Which of the following ordered pairs (x,y) satisfies the system of equations above?

   1. (3, -2)
   2. (2, -2)
   3. (-2, 2)
   4. (-2, -2)
   
   We will substitute an answer choice in each of the two equations.
   
   Try choice A):  x = 3 and y = -2
   First equation:   
   x +  y  =  0             3 -2 = 1, not 0         NO; choice A does not work.                                                                                                                             
   Now try choice B):  x = 2 and y = -2
   First equation:    x +  y  =  0                          2  -2  =  0          YES             
   Second equation:    3x -  2y  =  10        3(2) -2(-2)  =  6 + 4 = 10   YES   
   Thus our answer is B; answer choices C and D do not work.
   
   
2. x + 2y  =  7
   2x + 3y =  11
   Which of the following ordered pairs (x,y) satisfies the system of equations above?
   
   1. (-1, -3)
   2. (-1, 3)
   3. (1, 3)
   4. (1, -3)
   
   Try choice A):  x = -1 and y = -3
   First equation:   
   x +  2y  =  7             
   -1 +2(-3) = -1 -6  =  -7, not 7         
   NO; choice A does not work                                                                                                                                                                                             
   
   Try choice B):   x = -1 and y = 3
   First equation:   
   x +  2y  =  7             
   -1 +2(3) = -1 +6  =  5, not 7          NO; choice B does not work
   
   Try choice C):   x = 1 and y = 3
   First equation:   
   x +  2y  =  7             
   1 +2(3) = 1 +6  =  7    YES
   Second equation:   
   2x +  3y  =  11      2(1)  +  3(3)  =  2 + 9 = 11   YES                                      Thus our answer is C; answer choice D does not work.
   
   
3. -2x = 4y + 6
   -2x = 4y + 6                           
   2(2y + 3) = 3x - 5
   
   Which of the following ordered pairs (x,y) satisfies the system of equations above?
   
   1. (1, 2)
   2. (1, -2)
   3. (-1, -1)
   4. (-1, 1)
   
   

   Try choice A): x = 1 and y = 2
   First equation:
   -2x = 4y + 6
   -2x = -2(1) = -2 and 4y + 6 = 4(2) +6 = 8 + 6 = 14, not -2 NO; choice A does not work

   Now try choice B): x = 1 and y = -2
   First equation: -2x = 4y + 6 -2x = -2(1) = -2 and 4y + 6 = 4(-2) + 6 = -8 + 6 = -2 YES
   Second equation: 2(2y + 3) = 3x -5 2(2y + 3) = 2(2(-2) + 3) = 2(-4 + 3) = 2(-1) = -2 and 3x – 5 = 3(1) – 5 = 3 – 5 = -2 YES
   Thus our answer is B; answer choices C and D do not work.

SAT QUICK CHALLENGE EXERCISE K21
Transitions Within a Sentence Rather Than at the Beginning

The Relationship Between Two Sentences. An SAT question about an underlined transition enclosed in commas and located within a sentence tests your ability to identify the relationship between the underlined sentence and the sentence it follows. (For a quick refresher about selecting the right kind of transition, review the information on the TRANSITIONS MINI REVIEW CHART beneath the practice exercise below. Then, note Example A below.

Example A.  (1) Dr. Hall has enjoyed snacking on peanuts since she was a child.  (2) As an adult, for example, she gets a severe itch when she eats them.  In Sentence 2, “for example” says that the “severe itch” is an example of the enjoyment mentioned in Sentence 1.  However, we need a contrast transition that shows the contrast between “enjoyment” and a “severe itch.”  The word “however” in Example B, which follows, provides that contrast.  Example B.  Dr. Hall has enjoyed snacking on peanuts since she was a child.  As an adult, however, she gets a severe itch when she eats them. 

Combining Two Sentences. Sometimes an SAT transition question will ask you to identify the best way to combine two complete sentences. Note Example C, which follows. Example C. Directions. Select the best way to create a single sentence from the two sentences which follow. Combine the sentences at the underlined section where the two sentences meet.

During the COVID pandemic, many workers with strong computer skills have been able to work from home. Workers who can not use the computer to do their jobs have had to keep going to the office.

A. home, for, workers   B. home; and workers     C. home, whereas workers*   D. home moreover workers

Explanation of Example C: Since the two sentences express opposite ideas, a contrast transition is required. Therefore, choice C is correct. *Merging two sentences into one usually means placing a semicolon between the two (and starting the second one with a lower case letter), but when “whereas” begins the second sentence, a comma must be placed before “whereas,” and no punctuation should follow it. Now, keeping in mind the information in this lesson, complete the exercise below.

1. B
2. C
3. C

Equations of a Line Word Problems

SAT math often tests equation of a line concepts in the form of word problems. The questions will include phrases such as “at a constant rate” or “a flat fee.” In most cases you will create an equation in the form of y= mx + b, or interpret the components of an equation. The problems will usually start with a certain amount (a flat fee – meaning the y intercept) and increasing or decreasing at a constant rate (meaning the slope).

Now examine examples 1, 2, and 3.

1. The average number of students per classroom at Eastern High School from 2008 to 2021 can be modeled by the equation y = 0.38x + 24.5, where x represents the number of years since 2008, and y represents the average number of students per classroom. Which of the following best describes the meaning of the number 0.38 in the equation?

   1. The total number of students at the school in 2008
   2. The average number of students per classroom in 2008
   3. The estimated increase in the average number of students per classroom each year
   4. The estimated difference between the average number of students per classroom in 2021 and 2008
      

      The slope is 0.38, which means that for an increase of 1 in x, there is an increase of 0.38 in y, which is the average number of students per classroom. Thus the answer is C.

2. The equation y = 21.50 + 1.75x models the total cost y, in dollars, that a company charges a customer to rent a truck for one day and drive the truck x miles. The total cost consists of a flat fee plus a charge per mile driven. When the equation is graphed in the xy-plane, what does the y-intercept of the graph represent in terms of the model?
1. A flat fee of $21.50
2. A charge per mile of $1.75
3. A charge per mile of $21.50
4. Total daily charge of $23.25
   
   The y-intercept is the value of y when x is zero. In this problem it is the flat fee to which a charge per mile driven is added. Thus the answer is A.
   
   
3. To strengthen his abdominal muscles, Jerry has a 12 day plan to increase the number of sit-ups he can do each day. His plan calls for him to do 65 sit-ups by day 12 by increasing the number of sit-ups each day by a constant amount. If Jerry can do 20 sit-ups by day 3 and successfully complete his plan by day 12, how many sit-ups could he do before his plan was put into effect?
   1. 0
   2. 2
   3. 5
   4. 7

In terms of y = mx + b, we are asked to find the y-intercept, b. We can find b if we have values for y, m, and x; we need the equation of a line.

We have what can be considered to be coordinate points, (x₁,y₁) and (x₂,y2₎: day 12, 65 sit-ups and day 3, 20 sit-ups, (12,65) and (3,20). We can use these 3 steps to solve the problem:

1. Identify the coordinate points: (12,65) and (3,20).
2. Calculate the slope. The slope of a line can be determined from any two points on the line by using the slope formula: slope = (y₂ - y₁)/(x₂ - x₁) Slope = (65 – 20)/(12 – 3) = 45/9 = 5
3. Set up the equation of a line: y = mx + b. The slope, m, = 5. We can use either point for x and y, to solve for b. Using the point (12,65), we have 65 = 5(12) + b
   65 – 60 = b = 5
   
   

   Using the other point (3, 20), we have 20 = 5(3) + b
   20 – 15 = b = 5

   Thus, Jerry could do 5 sit-ups before he put his plan into effect. The answer is C.
   
   

   Keeping in mind the information above, solve the following problems.

1. A start-up company opened with 8 employees. The company’s growth plan assumes that 2 new employees will be hired each quarter (every three months) for the first 5 years. If an equation is written in the form of y = ax + b to represent the number of employees, y, employed by the company, what is the value of b?
   
   
   1. The approximate femur length, in inches, of a man with a height of 32.01 inches
   2. The approximate increase in a man’s femur length, in inches, for each increase of 32.01 inches in his height
   3. The approximate increase in a man’s femur length, in inches, for each one-inch increase in his height
   4. The approximate increase in a man’s height, in inches, for each one-inch increase in his femur length.The formula H = 1.88L + 32.01 can be used to approximate the height, H, in inches, of an adult male based on the length L, in inches, of his femur. What is the meaning of 1.88 in this context? 

2. Mr. Rollins is purchasing materials and renting tools needed for the concrete patio he is building in his back yard. The total cost, y, for buying the materials and renting the tools in terms of the number of days, x, is given by the equation y = M + (W + C)x where M is the materials cost in dollars, W is the rental cost of a wheelbarrow in dollars per day, and C is the rental cost of a concrete mixer in dollars per day. If this equation is graphed in the xy-plane, what does the slope of the line represent? 

   1. The total cost of the project
   2. The total cost of the materials
   3. The total daily cost of the project
   4. The total daily rental costs of the tools
   
   
   
   
3. The equation C = 75h + 125 gives the amount C, in dollars, an electrician charges for a job that takes h hours. Mrs. Cox and Mr. Lee each hired this electrician. The electrician worked two hours longer on Mr. Lee’s job than on Mrs. Cox’s job. How much more did the electrician charge Mr. Lee than Mrs. Cox?
   
   
   1. $75
   2. $125
   3. $150
   4. $200
   
   
   
4. A company purchased a machine valued at $120,000. The value of the machine depreciates by the same amount each year so that after 10 years the value will be $30,000. Which of the following equations gives the value, v, of the machine, in dollars, t years after it was purchased, where t is between 0 and 10 years?
   
   
   1. v = 30,000 – 9,000t
   2. v = 120,000 – 9,000t
   3. v = 120,000 + 9,000t
   4. v = 120,000 – 30,000t
   
   
   
5. A copy machine is initially loaded with 5,000 sheets of paper. The machine starts a large job and copies at a constant rate. After 20 minutes it has used 30% of the paper. Which of the following equations models the number of sheets of paper, p, remaining in the machine m minutes after the machine started printing?
   
   
   1. p = 5,000 – 20m
   2. p = 5,000 – 75m
   3. p = 5,000 – 0.3m
   4. p = 5,000 – 30m
   
   
   
6. The quantity of a product supplied in an economic market is a function of the price of the product. The estimated supply function is S = (1/2)P + 40 where S is the quantity of the product supplied to the market when the price is P dollars. How will the quantity supplied change if the price of the product is increased by $10?
   
   
   1. The quantity supplied will increase by 5 units
   2. The quantity supplied will decrease by 5 units
   3. The quantity supplied will increase by 10 units
   4. The quantity supplied will increase by 50 units
   
   
   

1. A start-up company opened with 8 employees. The company’s growth plan assumes that 2 new employees will be hired each quarter (every three months) for the first 5 years. If an equation is written in the form of y = ax + b to represent the number of employees, y, employed by the company, what is the value of b?
   
   b is the y-intercept, the amount we start with. Thus b is 8.
   
   
2. The formula H = 1.88L + 32.01 can be used to approximate the height, H, in inches, of an adult male based on the length L, in inches, of his femur. What is the meaning of 1.88 in this context? 
   
   1. The approximate femur length, in inches, of a man with a height of 32.01 inches
   2. The approximate increase in a man’s femur length, in inches, for each increase of 32.01 inches in his height
   3. The approximate increase in a man’s femur length, in inches, for each one-inch increase in his height
   4. The approximate increase in a man’s height, in inches, for each one-inch increase in his femur length.
   
   1.88 is the slope; it tells us what happens to H for a one unit increase in L. The answer is D.

3. Mr. Rollins is purchasing materials and renting tools needed for the concrete patio he is building in his back yard. The total cost, y, for buying the materials and renting the tools in terms of the number of days, x, is given by the equation y = M + (W + C)x where M is the materials cost in dollars, W is the rental cost of a wheelbarrow in dollars per day, and C is the rental cost of a concrete mixer in dollars per day. If this equation is graphed in the xy-plane, what does the slope of the line represent? 

   1. The total cost of the project
   2. The total cost of the materials
   3. The total daily cost of the project
   4. The total daily rental costs of the tools
   
   The slope is the total daily rental costs, not the total daily costs. The answer is D.
   
   
4. The equation C = 75h + 125 gives the amount C, in dollars, an electrician charges for a job that takes h hours. Mrs. Cox and Mr. Lee each hired this electrician. The electrician worked two hours longer on Mr. Lee’s job than on Mrs. Cox’s job. How much more did the electrician charge Mr. Lee than Mrs. Cox?
   
   1. $75
   2. $125
   3. $150
   4. $200
   
   The slope is $75; if h increases by 1, C increases by $75. If h increases by 2, C increases by 2($75) or $150. The answer is C.
   
   
5. A company purchased a machine valued at $120,000. The value of the machine depreciates by the same amount each year so that after 10 years the value will be $30,000. Which of the following equations gives the value, v, of the machine, in dollars, t years after it was purchased, where t is between 0 and 10 years?
   
   
   1. v = 30,000 – 9,000t
   2. v = 120,000 – 9,000t
   3. v = 120,000 + 9,000t
   4. v = 120,000 – 30,000t
   
   We need to determine the amount the machine depreciates each year.
   
   The amount of the decrease each year = ($120,000 - $30,000)/10 = $90,000/10 = $9,000. The slope is -$9,000; thus the answer is B since the value decreases each year.
   
   
6. A copy machine is initially loaded with 5,000 sheets of paper. The machine starts a large job and copies at a constant rate. After 20 minutes it has used 30% of the paper. Which of the following equations models the number of sheets of paper, p, remaining in the machine m minutes after the machine started printing?
   
   
   1. p = 5,000 – 20m
   2. p = 5,000 – 75m
   3. p = 5,000 – 0.3m
   4. p = 5,000 – 30m
   
   We need to determine how much is used each minute. 30% of the total amount is used in 20 minutes. 30% of 5,000 = 0.3(5,000) = 1,500 Thus 1,500 sheets are used in 20 minutes. The number of sheets used per minute = 1,500/20 = 75. Thus the number of sheets remaining equals the 5,000 we started with minus the number of sheets used in m minutes after the machine started printing : p = 5,000 – 75m. the answer is B.
   
   
7. The quantity of a product supplied in an economic market is a function of the price of the product. The estimated supply function is S = (1/2)P + 40 where S is the quantity of the product supplied to the market when the price is P dollars. How will the quantity supplied change if the price of the product is increased by $10?
   
   
   1. The quantity supplied will increase by 5 units
   2. The quantity supplied will decrease by 5 units
   3. The quantity supplied will increase by 10 units
   4. The quantity supplied will increase by 50 units
   
   The slope is ½. If the price increases by $1, the quantity supplied increases by ½ unit.
   If the price increases by $2, the quantity supplied increases by 1 unit.
   If the price increases by $10, the quantity supplied increases by 5 units.
   Thus the answer is A.

SAT QUICK CHALLENGE EXERCISE J21
Parallel Structures with Multiple Sentences

SAT parallelism questions often ask you to identify or create parallel structure within a single sentence, but some questions ask you to do that job when more than one sentence is involved. For those questions, the correct answer is always the choice that has the same pattern as the sentences which the underlined sentence comes after. Note Example A, which follows. Example A. Tell which choice best maintains the sentence pattern already established in the following paragraph.
Freezing rain made driving hazardous. Ice storms caused schools and businesses to close. It was heavy, wet snowflakes that turned the grass white.  

1. NO CHANGE
2. The grass was turned white by heavy, wet snowflakes.
3. Heavy, wet snowflakes turned the grass white.
4. The white grass was caused by heavy, wet snowflakes. 

Explanation of Example A. The first two sentences in the paragraph both begin with the adjective, noun, active (not passive) verb pattern. In fact, each of the first two sentences begins with an adjective phrase that is followed by an active verb, and then the rest of the sentence. The correct answer must follow that same pattern, and the correct answer is choice C.

Now, keeping in mind the information above, complete Exercise J21 below. Then use the answer key to check your work.

1. C
2. B
3. D

Equation of a Line

These are some points to remember about the equation of a line..:

1. The equation of a line is y = mx + b, where m is the slope of the line and b is the y-intercept.

2. The slope of a line shows the direction and steepness of the line. The slope can be positive or negative. If the slope is positive, the line is upward sloping. If the slope is negative, the line is downward sloping.
   
   The line is more vertical if the absolute value of the slope is large, and the line is more horizontal if
   the absolute value of the slope is small; at the limit, the slope of a horizontal line is zero.

3. The slope tells us what change takes place in y for a one unit increase is x. Consider the equation:
   y = 4x + 3 (the slope = 4)
   If x = 2, then y = 4(2) + 3 = 8 + 3 = 11
   If x increases by 1, then y increases by 4: y = 4(3) + 3 = 12 + 3 = 15 (an increase of 4)
   (The line is upward sloping; when x increases, y also increases.)

   Consider this equation: y = -3x + 6 (the slope = -3)
   If x = 4, then y = -3(4) + 6 = -12 + 6 = -6
   If x increases by 1, then y decreases by 3: y = -3(5) + 6 = -15 + 6 = - 9 (a decrease of 3)
   (The line is downward sloping; when x increases, y decreases.)

4. The slope of a line can be calculated from any two points on the line. The slope formula is:
   m = (y₂ – y₁)/(x₂ – x₁) where the two points are (x₁,y₁) and (x₂,y₂).
   If the two points are (3,6) and (5,10), the slope = (10 – 6)/(5 – 3) = 4/2 = 2
5. .The y-intercept is the value of y when the line crosses the y axis; it is the value of y when x = 0. In SAT problems, the y-intercept represents a base or starting amount before another amount is added. For example, you may be renting a car and the charge is a base amount plus an additional amount based on the number of miles driven.
   
   
6. There is also an x-intercept. The x-intercept is the value of x when the line crosses the x axis; it is the value of x when y = 0.
   
   
7. If a particular point is on a line, the equation of the line works out when that point is plugged into the equation for x and y. For example, to determine whether the point (2,10) is on the line y = 3x + 4, we substitute the x value, 2, into the equation, and we should get the y value, 10. In this case, the point (2,10) is on the line y = 3x + 4 since 3(2) + 4 = 6 + 4 = 10.
   But the point (4,5) is not on the line y = 3x + 4 since 3(4) + 4 = 12 + 4 = 16, not 5.

8. Parallel lines: two lines are parallel if their slopes are the same.

9. Perpendicular lines: two lines are perpendicular if the slope of one line is the negative reciprocal of the slope of the other line. For example, if the slope of line a is 2, the slope of a line perpendicular to line a is -1/2. If the slope of line j -4/5, the slope of a line perpendicular to line j is 5/4.

10. If we are given the slope of a line and a point on that line, we can determine the equation of the line. For example, if the slope of a line = 2 and a point on that line is (4,11), we can proceed as follows to determine the equation of the line:

    Write the equation of a line:                    y = mx + b
    From the point (4,11), we have values for x and y; we need to find a value for b; lets call it h,
    and then solve for h:                                y = mx + b
                                                                    11 = 2(4) + h now solve for h
                                                                    11 = 8 + h
                                                                      3 = h
    Thus the equation of the line is y = 2x + 3
    We can check our results by plugging in the x and y values from the point (4,11) into our equation:
    y = mx + b
    y = 2(4) + 3 = 8 + 3 = 11
    
    We can now find other points on the line. For example, if x = 5, y = mx + b = 2(5) + 3 = 10 + 3 = 13. Thus the point (5, 13) is on the line.

Keeping in mind the information above, solve the following problems.

1. Which of the following equations represents a line that is parallel to the line with equation y = -3x + 4?

   1. 6x + 2y = 15
   2. 3x – y = 7
   3. 2x – 3y = 6
   4. x + 3y = 1
      
      

2. The line with the equation (4/5)x + (1/3)y = 1 is graphed in the x,y-plane. What is the x coordinate of the x-intercept of the line?  

3. Which of the following is the equation of a line that is perpendicular to y + 3 = 3x -8?

   1. y + 3x = 9
   2. y – 3x = 10
   3. 9y -6x = 18
   4. 3y + x = 27

4. If a line has a slope of -2 and it passes through the point (-3,2), what is its y-intercept?

   1. 6
   2. 0
   3. -4
   4. -6

5. The line y = mx + 7, where m is a constant, is graphed in the xy-plane. If the line contains the point (2,13), what is the slope of the line?

   1. 2
   2. 3
   3. 5
   4. 7
      
      

6. The line y = kx + 4, where k is a constant, is graphed in the xy-plane. If the line contains the point (c,d), where c ≠ 0 and d ≠ 0, what is the slope of the line in terms of c and d?

   1. (d – 4)/c
   2. (c – 4)/d
   3. (4 – d)/c
   4. (4 – c)/d
      
      

7. A line has intercepts at (a,0) and (0,b) in the x,y-plane. If a + b = 0 and a ≠ b, which of the following is true about the slope of the line?

   1. It is positive.
   2. It is negative.
   3. It equals 0.
   4. It is undefined.
      
      

1. The l

   1. Which of the following equations represents a line that is parallel to the line with equation y = -3x + 4?

      1. 6x + 2y = 15
      2. 3x – y = 7
      3. 2x – 3y = 6
      4. x + 3y = 1
         
         
      The slope of y = -3x + 4 is -3. The line with a slope of -3 is parallel to this line. Check each
      answer choice; solve for y and determine which one has a slope of -3.
      
      
      1. 6x + 2y = 15
         2y = -6x + 15
         y = -3x + 15/2                 Yes

      2. 3x – y = 7
         – y = -3x + 7
         y = 3x – 7                         No

      3. 2x – 3y = 6
         – 3y = -2x + 6
         y = (2/3)x – 2                    No
         
         
      4. x + 3y = 1
         3y = -x + 1
         y = -(1/3)x + 1/3               No The answer is A.

   2. The line with the equation (4/5)x + (1/3)y = 1 is graphed in the x,y-plane. What is the x coordinate of the x-intercept of the line?  

      The x-intercept is the value of x when y = 0. Let y = 0 in the equation and solve for x:

      (4/5)x + (1/3)y = 1
      (4/5)x + (1/3)(0) = 1
      (4/5)x = 1
      x = 5/4

   3. Which of the following is the equation of a line that is perpendicular to y + 3 = 3x -8?

      1. y + 3x = 9
      2. y – 3x = 10
      3. 9y -6x = 18
      4. 3y + x = 27
         
         

      Solve the equation, y + 3 = 3x -8, for y: y = 3x -11

      The slope of y = 3x -11 is 3. The slope of a line that is perpendicular to this line is the negative reciprocal of 3, which is -1/3. Check each answer choice; solve for y and determine which one has a slope of -1/3.

      1. y + 3x = 9
         y = - 3x + 9                           No
         
         
      2. y – 3x = 10
         y = 3x + 10                            No
         
         

      3. 9y -6x = 18
         9y = 6x + 18
         y = (6/9)x + 2
         y = (2/3)x + 2                           No

      4. 3y + x = 27
         3y = - x + 27
         y = -(1/3)x + 9                          Yes The answer is D

   4. If a line has a slope of -2 and it passes through the point (-3,2), what is its y-intercept?

      1. 6
      2. 0
      3. -4
      4. -6
         
         

      The y-intercept is the value of y when x = 0. Let x = 0 in the equation and solve for y. But we must first determine the equation.

      Write the equation of a line: y = mx + b
      From the point (-3,2), we have values for x and y; we need to find a value for b; let's call it k,
      and then solve for k:                          y = mx + b
                                                                   2 = -2(-3) + h                    Now solve for h
                                                                   2 = 6 + h
                                                                 -4 = h
      Thus the equation of the line is y = -2x -4
      Now let x = 0 in the equation and solve for y: y = -2(0) -4
      y = -4                       The answer is C.

   5. The line y = mx + 7, where m is a constant, is graphed in the xy-plane. If the line contains the point (2,13), what is the slope of the line?

      1. 2
      2. 3
      3. 5
      4. 7
         
         

      Substitute the values of the point, x = 2 and y = 13, into the equation and solve for the slope, m.
      
      y = mx + 7
      13 = 2m + 7
      6 = 2m
      m = 3                        The answer is B.                        

   6. The line y = kx + 4, where k is a constant, is graphed in the xy-plane. If the line contains the point (c,d), where c ≠ 0 and d ≠ 0, what is the slope of the line in terms of c and d?

      1. (d – 4)/c
      2. (c – 4)/d
      3. (4 – d)/c
      4. (4 – c)/d

      We solve this problem the same way we solved number 5:
      Substitute the values of the point, c = x and d = y, into the equation and solve for the slope, k.

      y = kx + 4
      d = kc + 4
      d – 4 = kc
      (d – 4)/c = k              The answer is A.
      
      

   7. A line has intercepts at (a,0) and (0,b) in the x,y-plane. If a + b = 0 and a ≠ b, which of the following is true about the slope of the line?

      1. It is positive.
      2. It is negative.
      3. It equals 0.
      4. It is undefined.
         The slope = m = (y2 – y1)/(x2 – x1) where the two points are (a,0) and (0,b).
         m = (b – 0)/(0 – a) = b/-a = -b/a
         Since the slope = -b/a, many students would choose B as the answer. However, we need to know the sign of b/a. We have a negative times b/a. If b is positive and a is positive, then b/a is positive; if b is negative and a is negative, then b/a is positive; but if one if them is positive and the other one is negative, then b/a is negative. We are told that a + b = 0. Therefore one of them must be positive and the other one is negative. Thus a/b is negative, and we have m = - (b/a);
         and m equals a negative times a negative. Thus the slope is positive. The answer is A.

SAT QUICK CHALLENGE EXERCISE I21
The Pronoun and the Apostrophe 

What is the difference between a noun and a pronoun? Well, there are many, but this lesson focuses on one -- the use of the apostrophe. We add an apostrophe and the letter "s" to a noun to show ownership. For example, we write Rick's book or Ava's dentist, or even the people's choice.

The Pronoun and the Apostrophe. We never add an apostrophe and the letter "s" to a pronoun to show ownership. Instead, we use special pronoun ownership words. For Rick, we would write "his" book; for Ava, we would write "her" dentist; and for the people, we would write "their" choice.

Sometimes, we do use the apostrophe and the letter "s" for pronouns -- but only to make contractions.
You will recall that a contraction is a ''short cut;" it combines two words to make one word. Examples of contractions include "I'm" (I am), "You're" (You are), etc. Remembering the difference between how apostrophes are used for nouns and how they are used for pronouns will help you earn points each time an SAT question requires you to show that you know the difference.

Study the possessive pronouns, subject pronouns, and pronoun contractions in the chart below. Then, complete Exercise I21 (beneath the chart), and use the Answer Key to check your work.

1. C
2. B
3. C

Linear Equations: One Variable in Terms of Another

When solving linear equations, sometimes there is a need to solve for one variable in terms of another variable. For example, an SAT problem may ask you to solve for “a” in terms of “b”, or solve for “x” in terms of “y and z”. Some students who can readily solve linear equations where there is only one variable are not sure what to do when they are asked to solve a problem involving one variable in terms of another variable.

Actually, the process for determining one variable in terms of another variable is similar to solving an equation where there is only one variable. We isolate the variable for which we are solving on one side of the equal sign and put everything else on the other side of the equal sign, remembering that whatever we do to one side of the equation, we must do the same for the other side of the equation. We can perform several operations to both sides of the equation: add, subtract, multiply, divide, raise to a power, or take a root.
Now examine examples 1, 2, and 3.

Example 1. Solve for x in terms of m: 2x – 3 = m + x

                             Solve for x in terms of m                                       Similar problem with one variable
                                                2x – 3 = m + x                                      2x – 3 = 4 + x (m = 4)
Subtract x from both sides:     x – 3 = m                                              x – 3 = 4
Add 3 to both sides:                       x = m + 3                                  x = 4 + 3 = 7

Example 2. Solve for x in terms of y: (5 + 6x)/2 = y + x

Solve for x in terms of y                                                              Similar problem with one variable
(5 + 6x)/2 = y + x                                                                             (5 + 6x)/2 = 3 + x (y = 3)

Multiply both sides by 2:                     5 + 6x = 2(y + x)                                                            5 + 6x = 2(3 + x)
                                                               5 + 6x = 2y + 2x                                                           5 + 6x = 2(3) + 2x
Subtract 2x from both sides:                5 + 4x = 2y                                                                   5 + 4x = 2(3)
Subtract 5 from both sides:                        4x = 2y – 5                                                                   4x = 2(3) – 5
Divide both sides by 4:                                  x = (2y – 5)/4                                                                x = (2(3) – 5)/4 = (6 – 5)/4 = 1/4

Example 3. Solve for b in terms of a, c, and x: a(b - 2)/(c – 3) = x

                                              Solve for b in terms of a, c, and x                          Similar problem with one variable
                                                         a(b - 2)/(c – 3) = x                                           4(b - 2)/(7 – 3) = 9 (a = 4, c = 7, x = 9)
Multiply both sides by (c – 3)  :                     a(b-2) = x(c-3)                                              4(b-2) = 9(7 – 3)
Divide both sides by a:                                  b – 2 = x(c-3)/a                                               b – 2 = 9(7-3)/4
Add 2 to both sides:                                             b = x(c-3)/a + 2                                               b = 9(7-3)/4 + 2 = 9(4)/4 + 2 = 9+2=11

Keeping in mind the information above, solve the following problems.

1. Solve for x in terms of y: (x + 2y)/3 = 6 + 3x

2. Solve for a in terms of x and b: x = – b/2a

3. Solve for x in terms of a and n: (3ax – n)/5 = – 4

4. Solve for y in terms of b and c: (by + 2)/3 = c

5. Solve for x in terms of e, y, and z: ex – 2y = 3z

6. Solve for b in terms of k, m, and x: k = (x/2)(b + m)

1. Solve for x in terms of y: (x + 2y)/3 = 6 + 3x
   Multiply both sides by 3: x + 2y = 3(6 + 3x)
   x + 2y = 18 + 9x
   Subtract x from both sides: 2y = 18 + 8x
   Subtract 18 from both sides: 2y - 18 = 8x
   Divide both sides by 8: (2y – 18)/8 = x
   x = (2y – 18)/8 = (y – 9)/4

2. Solve for a in terms of x and b: x = – b/2a
   Multiply both sides by 2a: 2ax = – b
   Divide both sides by 2x: a = – b/2x

3. Solve for x in terms of a and n: (3ax – n)/5 = – 4
   Multiply both sides by 5: 3ax – n = – 20
   Add n to both sides: 3ax = n – 20
   Divide both sides by 3a: x = (n – 20)/3a

4. Solve for y in terms of b and c: (by + 2)/3 = c
   Multiply both sides by 3: by + 2 = 3c
   Subtract 2 from both sides: by = 3c – 2
   Divide both sides by b: y = (3c – 2)/b

5. Solve for x in terms of e, y, and z: ex – 2y = 3z
   Add 2y to both sides: ex = 3z + 2y
   Divide both sides by e: x = (3z + 2y)/e

6. Solve for b in terms of k, m, and x: k = (x/2)(b + m)
   Multiply both sides by 2: 2k = x(b + m)
   Divide both sides by x: 2k/x = b + m
   Subtract m from both sides: 2k/x – m = b

SAT QUICK CHALLENGE EXERCISE H21
Parallelism and Faulty Word Pairs Review

What Is Parallelism? When a sentence has two or more words, phrases, or clauses that have the same level of importance, maintain parallelism by using the same format for each of them. If you use a noun to identify one of the items, use a noun to identify each of the others. For instance, if Anna said, “I love to make ice cream and baking cakes,” the sentence would not be parallel because her sentence would have an infinitive (to make) and a gerund (baking). She could correct the sentence by saying “I love making ice cream and baking cakes.” (two gerunds) She could also say “I love to make ice cream and to bake cakes.” (two infinitives) However, since the infinitive “to” applies to both verbs, “make” and “bake,” the second “to” is optional and can be omitted for the sake of conciseness.

Word Pairs and Comparisons. Some SAT questions ask about word pairs noted for pointing out similarities and differences such as “either…or, neither…nor,” or not only….but also. Always use the pairs together; do not mix or match them with any other words. Also, be sure that the words or phrases that follow the parts of a word pair are parallel. Note the examples in the chart below.

Practice Exercise

Directions. For each question below, write the letter of the answer choice which corrects the underlined part of the sentence. If the underlined part is already correct, select choice A (NO CHANGE). Use the answer key beneath the questions to check your work.

1. C
2. A
3. B

Linear Equations: Additional Word Problems

The lesson last week dealt with word problems that could be solved by setting up an equation. Because of the large number of word problems on the SAT, additional word problems will be reviewed this week. It was noted that the following steps can help solve word problems:

1. Read the problem carefully and avoid misreading anything important.
2. Identify the key values and determine how they are related mathematically by converting the words to math.
   1. It is necessary to define the variables and create equations to represent relationships. State in words what the unknown is.
   2. Carefully determine the equality: identify what goes on the left side of the equal sign and what goes on the right side of the equal sign.
3. Solve the problem and interpret the solution.
4. Make sure that the answer is reasonable. Ask yourself: Does it make sense? Does it answer the question that was asked?
5. Warning: beware of a possible change in units in the problem. Information in the problem may be in one unit while the question to be answered may be in another unit. For example, information in the problem may be in feet, but the question to be answered may be in inches.

Solve the following problems

1. A website hosting service charges businesses a one time setup fee of $350 plus a monthly amount. If a business paid $1,010 for the first 12 months, including the setup fee, what was the monthly amount?
   1. $25
   2. $35
   3. $45
   4. $55
      
      
2. The Andersons, a retired couple, made two trips to Europe last year. The first trip lasted 15 days longer than the second trip, and the two trips combined lasted a total of 287 days. How many days did the second trip last? 
   1. 122
   2. 136
   3. 146
   4. 155
      
      
3. Marsha made a deposit into her investment account on July 1, 2011. The amount of money in the account doubled each year until Marsha had $2,400 in her investment account on July 1, 2015. What was the amount of the deposit made on July 1, 2011?
   
   1. 150
   2. 800
   3. 1,000
   4. 1,200
      
      
4. Six years ago, I was three-fourths as old as I will be 7 years from now. How old am I now??
   1. 32
   2. 38
   3. 45
   4. 53
      
      
5. The sum of the ages of three children is 29. The age of the oldest is three times the age of the youngest. The two oldest children differ in age by 6 years. What is the age of the youngest child?
   
   1. 5
   2. 7
   3. 10
   4. 16
      
      
6. The Zylox Corporation produces several saline solutions that have a variety of medical and industrial uses. How many gallons of a 25% saline solution must be added to 40 gallons of a 10% saline solution to obtain a 15% saline solution?
   1. 9
   2. 14
   3. 20
   4. 25
      
      
7. Charles drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Charles would like to reduce his weekly expenditure on gasoline by $5. If gasoline costs $4 per gallon, how many fewer average miles should he drive each week?
   
   1. 23.25
   2. 31.25
   3. 52.75
   4. 68.75

1. 1. The variable is the monthly amount.
   2. The amount paid for the first 12 months, 1010, goes on the right side of the equal sign.
   3. The one time setup fee, 350, plus the monthly payments, 12x, goes on the left side of the equal sign.
      
      350 + 12x = 1010.
      12x = 660
      x = 55 -------- D      The monthly amount is $55.
      A website hosting service charges businesses a one time setup fee of $350 plus a monthly amount. If a business paid $1,010 for the first 12 months, including the setup fee, what was the monthly amount?
      
      1. $25
      2. $35
      3. $45
      4. $55
      
      To answer this question, we need to determine the elements of the equation: what the variable is, what goes on the right side of the equal sign, and what goes on the left side of the equation.
      
      
2. The Andersons, a retired couple, made two trips to Europe last year. The first trip lasted 15 days longer than the second trip, and the two trips combined lasted a total of 287 days. How many days did the second trip last?
   
   1. 122
   2. 136
   3. 146
   4. 155
   
   The variable is the length of the second trip.
   The combined length of the two trips, 287 days, goes on the right side of the equal sign.
   The length of the second trip, x, plus the length of the first trip, 15 + x, goes on the left side of the equal sign.
   
   x + 15 + x = 287
               2x = 272
                 x = 136 -------- B         The second trip lasted 136 days.
   
   
3. Marsha made a deposit into her investment account on July 1, 2011. The amount of money in the account doubled each year until Marsha had $2,400 in her investment account on July 1, 2015. What was the amount of the deposit made on July 1, 2011?
   
   1. 150
   2. 800
   3. 1,000
   4. 1,200
   
   The variable is the amount deposited on July 1, 2011.
   The amount in the account on July 1, 2015, $2,400, goes on the right side of the equal sign.
   2⁴x, goes on the left side of the equal sign. The “2” indicates that the amount doubled each year; the
   “4” indicates that there are 4 years between July 1, 2011 and July 1, 2015.
   
   

   2⁴x = 2400
   16x = 2400
   x = 150 ----------- A           The amount deposited on July 1, 2011, was $150.

   If “2⁴x = 2400” is not obvious, we could also solve this problem as follows, noting that the amount doubled each year:

   Amount on July 1, 2011 = x
   Amount on July 1, 2012 = 2x
   Amount on July 1, 2013 = 4x
   Amount on July 1, 2014 = 8x
   Amount on July 1, 2015 = 16x

   16x = 2,400
       x = 150

4. Six years ago, I was three-fourths as old as I will be 7 years from now. How old am I now?
   
   1. 32
   2. 38
   3. 45
   4. 53
   
   The variable is my age now.
   ¾ of my age 7 years from now, ¾(x + 7), goes on the right side of the equal sign.
   My age 6 years ago, (x – 6), goes on the left side of the equal sign.
   
     (x – 6) = ¾(x + 7)
   4x – 24 = 3x + 21
              x = 45 ---------- C                 My current age is 45. 
   
   
5. The sum of the ages of three children is 29. The age of the oldest is three times the age of the youngest. The two oldest children differ in age by 6 years. What is the age of the youngest child?
   
   1. 5
   2. 7
   3. 10
   4. 16

   The variable is the age of the youngest child.
   The sum of the ages, 29, goes on the right side of the equal sign.
   The age of the youngest child + the age of the middle child + the age of the oldest child goes on the left side of the equal sign.

   The age of the youngest child = x
   The age of the oldest child = three times the youngest = 3x
   The age of the middle child is 6 less than the age of the oldest child = 3x – 6

   x + 3x + 3x – 6 = 29
                 7x – 6 = 29
                       7x = 35
                         x = 5 ---------- A               The youngest child is 5, the oldest child is 15, and the
                                                                     middle child is 9.

6. The Zylox Corporation produces several saline solutions that have a variety of medical and industrial uses. How many gallons of a 25% saline solution must be added to 40 gallons of a 10% saline solution to obtain a 15% saline solution?
   
   1. 9
   2. 14
   3. 20
   4. 25

   The variable is the number of gallons of the 25% saline solution.
   The 15% solution, 0.15(40 + x), goes on the right side of the equal sign.
   The 10% solution + the 25% solution, 0.10(40) + 0.25x, goes on the left side of the equal sign.

   0.10(40) + 0.25x = 0.15(40 + x)
              4 + 0.25x = 6 + 0.15x
                    0.10x = 2
                           x = 20 ----------- C 20 gallons of the 25% saline solution must be added.

7. Charles drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Charles would like to reduce his weekly expenditure on gasoline by $5. If gasoline costs $4 per gallon, how many fewer average miles should he drive each week?
   
   1. 23.25
   2. 31.25
   3. 52.75
   4. 68.75
   
   The variable is the number of miles driven to result in a $5 reduction in weekly expenditure.
   
   The amount of weekly saving, $5, goes on the right side of the equal sign.
   
   The weekly amount spent now, (100/25)(4), minus the weekly amount spent to result in a $5 reduction in expenditure (x/25)(4), goes on the left side of the equal sign.
   
   (100/25)(4) - (x/25)(4) = 5
                 (4)(4) – 4x/25 = 5
                     16 – 4x/25 = 5
                        400 – 4x = 125
                                – 4x = – 275
                        x = 68.75 = weekly miles needed to drive to reduce weekly expenditure by $5
   
       100 – 68.75 = 31.25 miles = fewer average miles he should drive each week -------- B

SAT QUICK CHALLENGE EXERCISE G1
Correcting Modifier Errors

Putting the Modifier in the Right Place. A modifier describes or provides additional information about someone or something. A modifier that is not placed next to (or as close as possible to) the word/word group it should describe is misplaced, and the sentence it is in does not convey the meaning intended. Note the underlined, misplaced modifier in Example A, which follows. Example A: Thoughtless telemarketers really annoy homeowners making calls early in the morning or late at night. Explanation. Since the modifier is next to the word “homeowners,” the sentence says that homeowners are making calls early in the morning or late at night. However, the modifier is intended to describe the telemarketers. Note the correction in Example B, which follows. Example B: Thoughtless telemarketers making calls early in the morning or late at night really annoy homeowners. Explanation. Now, the modifier is next to the word “telemarketers,” and the sentence says that “telemarketers” are making calls early in the morning or late at night. Therefore, the sentence is written correctly.

The Introductory Modifying Phrase Error. “Misplaced modifier” is a general phrase that applies to any modifier that is not in the right place. However, sometimes more specific terminology is used to indicate exactly what kind of modifier error has been made. For instance, note the introductory modifying phrase error in Example C, which follows: Example C: Completely exhausted after working overtime for six hours, the bed was a very welcome sight to Lori. Explanation. Because the modifier at the beginning of the sentence is closest to the noun “bed,” the sentence says that the bed was completely exhausted after it had worked overtime for six hours. Of course, that sentence does not make sense. Lori,-- not the bed -- had worked overtime. Now, note Example D, which follows: Example D: Completely exhausted after working overtime for six hours, Lori found the bed to be a very welcome sight. Explanation. Since the modifier is right beside the word it should modify, “Lori,” the modifier is in the right place, and the sentence makes sense. Now, keeping in mind the information above, complete the exercise below.
 

1. D
2. C
3. B

Linear Equations: Word Problems

An increasing number of problems on the SAT are word problems. The following steps can help you solve word problems:

1. Read the problem carefully and avoid misreading anything important.
2. Identify the key values and determine how they are related mathematically by converting the words to math.  
   1. It is necessary to define the variables and create equations to represent relationships.  State in words what the unknown is.  
   2. Carefully determine the equality:  identify what goes on the left side of the equal sign and what goes on the right side of the equal sign.
3. Solve the problem and interpret the solution.
4. Make sure that the answer is reasonable.  Ask yourself:  Does it make sense?  Does it answer the question that was asked?
5. Warning:  beware of a possible change in units in the problem.  Information in the problem may be in one unit while the question to be answered may be in another unit.  For example, information in the problem may be in feet, but the question to be answered may be in inches.

Now examine examples 1, 2, and 3.

Example 1:  Betty and her younger sister Lynn work together to peel potatoes.  Betty peels 15 more than twice the number of potatoes that Lynn peels.  Together they peel 75 potatoes.  How many potatoes does Lynn peel?   
To answer this question, we need to determine the elements of the equation:  what the variable is, what goes on the right side of the equal sign, and what goes on the left side of the equation.  

1. The variable is the number of potatoes that Lynn peels.
2. The number of potatoes peeled, 75, goes on the right side of the equal sign.
3. The number of potatoes Betty peels, 2x + 15,  plus the number of potatoes Lynn peels, x,  goes on the left side of the equal sign.     
   2x + 15 + x  =  75.
   3x  =  60
     x  =  20            Lynn peels 20 potatoes.    

Example 2: Martin bought a laptop computer at a store that gave a 10% discount off the original price. The total amount he paid to the cashier was $715.50, which included a 6% sales tax that was added to the discounted price. What was the original price of the computer?

1. The variable is the original price of the computer.
2. The amount paid to the cashier, $715.50, goes on the right side of the equal sign.
3. The discounted price of the computer, (1 – 0.1)x times the sales tax adjustment, (1-0.1)x(1.06), goes on the left side of the equal sign.
            (1 – 0.1)x(1.06) = 715.50
                     (.9)x(1.06) = 715.50
                            .954x = 715.5
                                    x = 750                    The original cost of the computer was $750.00.

Example 3:  Oliver is selling photographs as part of a project for his entrepreneurship class.  He sells the first 20 photographs for $10 each.  Because the first 20 photographs sold so quickly, he raised the price of the photographs to $15 each for the rest of the project.  After his expenses, Oliver earns a profit of 80% of the revenues from his sales.  How many photographs must he sell for the rest of the project to earn a profit of $400?

1. 18
2. 20
3. 24
4. 32

The variable is the number of additional photographs he must sell.  The profit of $400 goes on the right side of the equal sign.  The total revenues times 80% goes on the left side of the equal sign.
The total revenues  =  $10 times 20  + ( $15 times x).  Thus we have:

0.8(200 + 15x)  =  400
       160 + 12x  =  400
                  12x  =  240
                        x  =  20  ------------ B

Keeping in mind the information above, answer the following questions.

1. A customer paid $81.00 for a coat after an 8% sales tax was added. What was the price of the coat before the sales tax was added?

   1. $72.00
   2. $74.52
   3. $75.00
   4. $87.48
      
      

2.  The weight limit for the contents in a paper bag is 25 pounds. A customer is purchasing a three pound cantaloupe and several boxes of spaghetti each weighing 24 ounces. What is the maximum number of boxes of spaghetti that can be put in the paper bag so that the weight of the contents is below the maximum weight limit?

3. The cost of using a telephone in a hotel meeting room is $0.15 per minute. Which of the following equations represents the total cost C, in dollars, for x hours of phone use? 

1. 0.15x  =  C
2. 0.15(60x)  =  C
3. 60x/0.15  =  C
4. 0.15x/60  =  C
   
   

4. A school district is forming a committee to discuss plans for the construction of a new high school. Of those invited to join the committee, 15% are parents of students, 45% are teachers from the current high school, 25% are school and district administrators, and 6 individuals are students. How many more teachers were invited to join the committee than school and district administrators?

5. The mean score of 8 players in a basketball game was 14.5 points. If the highest individual score is removed, the mean score of the remaining 7 players becomes 12 points. What was the highest score?

   1. 20
   2. 24
   3. 32
   4. 36
      
      

6. Max is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Max saves 90% of his earnings from each week. How many hours must he work the rest of the week to save $270 for the week?

   1. 16
   2. 22
   3. 33
   4. 38

1. A customer paid $81.00 for a coat after an 8% sales tax was added. What was the price of the coat before the sales tax was added?
   
   1. $72.00
   2. $74.52
   3. $75.00
   4. $87.48
   
   The variable is the price of the coat before the sales tax is added. The amount the customer paid of $81.00 goes on the right side of the equal sign. The initial price plus the sales tax goes on the left side of the equal sign.
       x + .08x = 81
           1.08x = 81
                   x = 75 ----------- C
   
   

2. The weight limit for the contents in a paper bag is 25 pounds. A customer is purchasing a three pound cantaloupe and several boxes of spaghetti each weighing 24 ounces. What is the maximum number of boxes of spaghetti that can be put in the paper bag so that the weight of the contents is below the maximum weight limit?
   
   The variable is the number of boxes of spaghetti. The maximum weight of the contents that can be put in the paper bag, 25 pounds, goes on the right side of the equal sign. There is an issue of units in this problem; every item should be expressed in pounds or ounces. In this case we will express everything in pounds. The weight of the contents in the paper bag goes on the left side of the equal sign.
   
   Since we are expressing each item in pounds, change the weight of the spaghetti boxes from 24 ounces to 1.5 pounds.
   
             3 + 1.5x = 25
                   1.5x = 22
                        x = 14.67           The maximum number of boxes of spaghetti is 14.
   
   If we express the weight of each item in ounces, we have 3 pounds = 48 ounces and 25 pounds = 25 times 16 = 400 ounces. Thus our equation is:
   
   48 + 24x = 400
           24x = 352
               x = 14.67 The maximum number of boxes of spaghetti is 14.

3. The cost of using a telephone in a hotel meeting room is $0.15 per minute. Which of the following equations represents the total cost C, in dollars, for x hours of phone use?
   
   1. 0.15x = C
   2. 0.15(60x) = C
   3. 60x/0.15 = C
   4. 0.15x/60 = C
   
   Since the hotel charges $0.15 per minute, we convert the per minute rate to an hourly rate: $0.15 per minute times 60 minutes per hour = 0.15(60) per hour. Then we multiply by the number of hours of phone use. Thus the equation is: 0.15(60x) = C Choice B is correct.
   
   

4. A school district is forming a committee to discuss plans for the construction of a new high school. Of those invited to join the committee, 15% are parents of students, 45% are teachers from the current high school, 25% are school and district administrators, and 6 individuals are students. How many more teachers were invited to join the committee than school and district administrators?
   
   The variable is the number of people invited to join the committee. The total number of people invited to join the committee goes on the right side of the equal sign. The total of the individual groups goes on the left side of the equal sign.
   
   Thus our equation is:                    0.15x + 0.45x + 0.25x + 6 = x
                                                                                     0.85x + 6 = x
                                                                                                  6 = 0.15x
                                                                                                   x = 6/0.15 = 40
   
   40 people were invited to join the committee.
   
   Teachers:                                                                                 0.45(40) = 18
   School and district administrators:                                      0.25(40)  = 10
                                                                                                                      --
                                                                                                                        8 Our answer is 8.

5. The mean score of 8 players in a basketball game was 14.5 points. If the highest individual score is removed, the mean score of the remaining 7 players becomes 12 points. What was the highest individual score?
   
   1. 20
   2. 24
   3. 32
   4. 36
   
   

   The variable is the highest individual score. The total number of points scored by the
   remaining 7 players goes on the right side of the equal sign. The total number of
   points sored by all 8 players minus the highest individual score goes on the left side
   of the equal sign.

   Remember that in a mean (average) problem the first thing to do is to find the total. There are two ways to find the total: 1) add all of the items; 2) multiply the mean by the number of items. On the SAT the second method is used more often.
   
   The total of the seven players = 7(12) = 84
   The total of the eight players = 8(14.5) = 116
   
   The equation is: 116 – x = 84
                                        -x = -32
                                          x = 32 --------- C

6. Max is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Max saves 90% of his earnings from each week. How many hours must he work the rest of the week to save $270 for the week?
   
   1. 16
   2. 22
   3. 33
   4. 38
   
   The variable is the number of additional hours he must work. The savings of $270 goes on the right side of the equal sign. The total earnings times 90% goes on the left side of the equal sign. The total earnings = $8 times 10 + $10 times x. Thus we have: 0.9(80 + 10x) = 270
                                                                             72 + 9x = 270
                                                                                     9x = 198
                                                                                       x = 22 -------------  B