ACT Math : Equations / Inequalities

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #241 : Equations / Inequalities

Solve  and .  What is the sum of  and ?

Possible Answers:

Correct answer:

Explanation:

Adding the two equations together gives , so .  Substituting  into one of the original equations gives

The sum of  and  is

Example Question #23 : Systems Of Equations

Suppose x2 + x – 6 = 0. Which of the following could be a value of x?

Possible Answers:

2

3

1

0

4

Correct answer:

2

Explanation:

Factor out this binomial. –3 and 2 are the only possible x values. 2 is the answer.

Example Question #22 : Systems Of Equations

Find a solution for the following system of equations:

x – = 2

2x – 3y = 3

Possible Answers:

(0, 0)

(0, 1)

(1, 3)

(3, 0)

(3, 1)

Correct answer:

(3, 1)

Explanation:

Use substitution and solve for one variable, then back substitute and solve for the other variable, or use elimination. 

Example Question #23 : Systems Of Equations

Solve the following system of equations:

Possible Answers:

Correct answer:

Explanation:

There are two ways to solve this problem.

Option 1: The Substitution Method

Step 1: Set up the second equation so that  is by itself

Step 2: Substitute for  in the first equation, and solve for 

Step 3: Plug  into the second equation and solve for 

Option 2: The Elimination Method

Step 1: Set up the equations so that the variables are on the same side

Step 2: Multiple the second equation by 2

Step 3: Subtract the second equation from the first (thereby canceling out the s) and solve for x

________________

Step 4: Substitute  into one of the equations and solve for 

Example Question #31 : Systems Of Equations

Solve for  and .

Possible Answers:

Correct answer:

Explanation:

Setting both equations equal to  gives

and

Setting these expressions equal to each other gives

So, .

Plugging that back into the first equation:

The final answer is  and .

Example Question #1 : How To Find The Solution For A System Of Equations

Solve for .

Possible Answers:

Correct answer:

Explanation:

For the second equation, solve for  in terms of .

Plug this value of y into the first equation.

Example Question #44 : How To Find The Solution For A System Of Equations

A store sells 17 coffee mugs for $169. Some of the mugs are $12 each and some are $7 each. How many $7 coffee mugs were sold?

Possible Answers:

7

10

8

6

9

Correct answer:

7

Explanation:

The answer is 7. 

Write two independent equations that represent the problem. 

x + y = 17 and 12x + 7y = 169

If we solve the first equation for x, we get x = 17 – y and we can plug this into the second equation. 

12(17 – y) + 7y = 169

204 – 12y + 7y =169

–5y = –35

y = 7

Example Question #1 : Linear Equations With Whole Numbers

What is the solution of  for the systems of equations?

Possible Answers:

Correct answer:

Explanation:

We add the two systems of equations:

For the Left Hand Side:

For the Right Hand Side:

So our resulting equation is:

 

Divide both sides by 10:

For the Left Hand Side:

For the Right Hand Side:

Our result is:

Example Question #45 : How To Find The Solution For A System Of Equations

What is the solution of  that satisfies both equations?

Possible Answers:

Correct answer:

Explanation:

Reduce the second system by dividing by 3.

Second Equation:

     We this by 3.

Then we subtract the first equation from our new equation.

First Equation:

First Equation - Second Equation:

Left Hand Side:

Right Hand Side:

Our result is:

Example Question #2 : Linear Equations With Whole Numbers

What is the solution of  for the two systems of equations?

Possible Answers:

Correct answer:

Explanation:

We first add both systems of equations.

Left Hand Side:

Right Hand Side:

Our resulting equation is:

 

We divide both sides by 3.

Left Hand Side:

Right Hand Side:

Our resulting equation is:

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