SAT Mathematics : Solving Linear Equations

Study concepts, example questions & explanations for SAT Mathematics

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Example Questions

Example Question #1 : Solving Linear Equations

If , what is the value of ?

Possible Answers:

13

9

19

23

Correct answer:

19

Explanation:

The first step on this problem is to take the equation you're given, , and solve for . To do that, divide both sides by  to arrive at . Importantly, recognize that the question is not asking for the value of , but rather for the value of the expression . To find that answer, plug in  and the expression will look like . This then equals , making the correct answer .

Example Question #2 : Solving Linear Equations

If , then what is the value of ?

Possible Answers:

0.25

12

4

36

Correct answer:

0.25

Explanation:

Whenever you're solving for a variable in an equation that has fractions, it's a good first step to multiply both sides by the denominator to get all the variables and values outside of the fractions. Here that means. multiplying both sides by  to get .

Next, you have a linear equation in which the variable has a coefficient, so divide by the coefficient to isolate the variable. If you divide both sides by , you then have: . You can then reduce the fraction  to , and since the answer choices only use integers and decimals, you can convert the fraction to the proper decimal, 

Example Question #3 : Solving Linear Equations

If , what is the value of ?

Possible Answers:

3

14

7

11

Correct answer:

11

Explanation:

The first step on this problem is to use the equation that you're provided so that you can solve for . To do so, take  and divide both sides by . That leaves you with . Of course, the question doesn't ask you for the value of , but rather the value of . So plug in  and the expression looks like:  which is . That simplifies to , giving you your final answer.

Example Question #4 : Solving Linear Equations

If , what is the value of  ?

Possible Answers:

Correct answer:

Explanation:

Note that this problem does not ask you to solve for a particular variable, but instead a particular fraction or ratio, . For that reason, your goal as you perform algebra should be to isolate the ratio of  and . You can do that by multiplying both sides by . That gives you:

, which simplifies to .  

You can then just flip each fraction (in doing so, you're doing the same thing to both sides, namely taking the reciprocal), to get:

, which simplifies to .

Example Question #5 : Solving Linear Equations

If  and , what is the value of ?

Possible Answers:

6

-6

0

12

Correct answer:

12

Explanation:

Whenever the SAT asks you to solve for the value of a variable, you have two options: you can either use your answers as assets and backsolve, or you can solve for the variable directly. In this case, both are equally good options, and you should use whichever method is most comfortable. Generally, however, solving directly is usually the best option on the SAT non-calculator section.

If you substitute the value  into the equation given, it becomes:

Then you can decide whether to solve directly or plug in the answer choices. If you solve directly, then multiply both sides by  to eliminate the denominator, yielding:

Your final step is then to divide both sides by  and you have .

If you were to plug in the answer choices, you would find that only  would work, as .

Example Question #6 : Solving Linear Equations

If , what is the value of ?

Possible Answers:

-1

-3

4

2

Correct answer:

-3

Explanation:

The first key to this problem is to combine like terms; each side of the equation has multiple  terms for you to sum to simplify the equation.  Your streamlined equation should look like:

Now you can get all the  terms on one side and all the numeric terms on the other so that you're ready to solve. That means subtracting  and  from each side, yielding:

Then you can divide both sides by  to arrive at your final answer:

Example Question #7 : Solving Linear Equations

If , then what is the value of ?

Possible Answers:

4

5

3

2

Correct answer:

4

Explanation:

To solve this problem algebraically, take the given equation and subtract  from both sides to isolate the  term. That gives you:

Then you can divide both sides by  to yield:

When you plug in  to , that gives you , which is .

Example Question #8 : Solving Linear Equations

What is the value of  in the equation above?

Possible Answers:

3

8

5

0

Correct answer:

5

Explanation:

To solve this problem algebraically, distribute the multiplication across each set of parentheses, remembering to multiply each term within the parentheses by its coefficient. That gives you:

Then you can combine like terms on the left side, since you have two  terms and two numeric terms. That simplifies to:

From there, subtract  from both sides and you have your answer, .

 

Example Question #8 : Solving Linear Equations

If , what is the value of ?

Possible Answers:

21

17

29

33

Correct answer:

29

Explanation:

To solve this problem, first perform algebra on the given equation to isolate the  terms on one side of the equation and the numeric terms on the other. That means subtracting  from both sides and adding  to both sides to get:

You can then divide both sides by  to realize that .

Now, notice that the question did not ask you for the value of , but rather for the value of . So you now need to plug in  for  to finish the job: 

Example Question #9 : Solving Linear Equations

If , then what is the value of ?

Possible Answers:

171

141

129

159

Correct answer:

159

Explanation:

To answer this problem, you need to first solve for  in the given equation. You can first multiply each side by  to eliminate the denominator, yielding:

Then divide each side by  to isolate :

Now you can plug in  for  in the given expression. That yields: 

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