### All ACT Math Resources

## Example Questions

### Example Question #241 : Equations / Inequalities

Solve and . What is the sum of and ?

**Possible Answers:**

**Correct answer:**

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 x^{2} + *x* – 6 = 0. Which of the following could be a value of *x*?

**Possible Answers:**

2

3

1

0

4

**Correct answer:**

2

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* – *y *= 2

2*x* – 3*y* = 3

**Possible Answers:**

(0, 0)

(0, 1)

(1, 3)

(3, 0)

(3, 1)

**Correct answer:**

(3, 1)

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:**

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:**

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:**

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

The answer is 7.

Write two independent equations that represent the problem.

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

204 – 12*y* + 7*y* =169

–5*y* = –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:**

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:**

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:**

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: