# ACT Math : Equations / Inequalities

## Example Questions

### Example Question #241 : Equations / Inequalities

Solve  and .  What is the sum of  and ?

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?

2

3

1

0

4

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

(0, 0)

(0, 1)

(1, 3)

(3, 0)

(3, 1)

(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:

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 .

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 .

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?

7

10

8

6

9

7

Explanation:

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?

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?

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?

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: