# Solving Linear Equations and Inequalities

## Solve Equations using the Division and Multiplication Properties of Equality


### Solve Equations Using the Division and Multiplication Properties of Equality

You may have noticed that all of the equations we have solved so far have been of the form  or . We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.
Let’s look at our puzzle again with the envelopes and counters in .
In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?
How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since ).
What equation models the situation shown in ? There are two envelopes, and each contains  counters. Together, the two envelopes must contain a total of 6 counters.
We found that each envelope contains 3 counters. Does this check? We know , so it works! Three counters in each of two envelopes does equal six!
This example leads to the Division Property of Equality.
The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.
Consider the equation . We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.
In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.

### Solve Equations That Require Simplification

Many equations start out more complicated than the ones we have been working with.
With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.
Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

### Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

### Translate and Solve Applications

To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.

### Key Concepts

1. The Division Property of Equality—For any numbers a, b, and c, and , if , then .
When you divide both sides of an equation by any non-zero number, you still have equality.
2. The Multiplication Property of Equality—For any numbers a, b, and c, if , then .
If you multiply both sides of an equation by the same number, you still have equality.

### Practice Makes Perfect

Solve Equations Using the Division and Multiplication Properties of Equality
In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.
Solve Equations That Require Simplification
In the following exercises, solve each equation requiring simplification.
Mixed Practice
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?