Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same – so that we end up with a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!

Solve Equations Using the Subtraction and Addition Properties of Equality

We are going to use a model to clarify the process of solving an equation. An envelope represents the variable – since its contents are unknown – and each counter represents one. We will set out one envelope and some counters on our workspace, as shown in Figure \(\PageIndex{1}\). Both sides of the workspace have the same number of counters, but some counters are “hidden” in the envelope. Can you tell how many counters are in the envelope?

What are you thinking? What steps are you taking in your mind to figure out how many counters are in the envelope?

Perhaps you are thinking: “I need to remove the 3 counters at the bottom left to get the envelope by itself. The 3 counters on the left can be matched with 3 on the right and so I can take them away from both sides. That leaves five on the right—so there must be 5 counters in the envelope.” See Figure \(\PageIndex{2}\) for an illustration of this process.

What algebraic equation would match this situation? In Figure \(\PageIndex{3}\) each side of the workspace represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope x.

Let’s write algebraically the steps we took to discover how many counters were in the envelope:

First, we took away three from each side.

Then we were left with five.

Table \(\PageIndex{1}\)

Check:

Five in the envelope plus three more does equal eight!

\[5+3=8\]

Our model has given us an idea of what we need to do to solve one kind of equation. The goal is to isolate the variable by itself on one side of the equation. To solve equations such as these mathematically, we use the Subtraction Property of Equality.

Let’s see how to use this property to solve an equation. Remember, the goal is to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.

What happens when an equation has a number subtracted from the variable, as in the equation \(x−5=8\)? We use another property of equations to solve equations where a number is subtracted from the variable. We want to isolate the variable, so to ‘undo’ the subtraction we will add the number to both sides. We use the Addition Property of Equality.

In Exercise \(\PageIndex{4}\), 37 was added to the y and so we subtracted 37 to ‘undo’ the addition. In Exercise \(\PageIndex{7}\), we will need to ‘undo’ subtraction by using the Addition Property of Equality.

The next example will be an equation with decimals.

Solve Equations That Require Simplification

In the previous examples, we were able to isolate the variable with just one operation. Most of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality.

You should always simplify as much as possible before you try to isolate the variable. Remember that to simplify an expression means to do all the operations in the expression. Simplify one side of the equation at a time. Note that simplification is different from the process used to solve an equation in which we apply an operation to both sides.

Translate to an Equation and Solve

To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign. Here are some of the words that are commonly used.

Equals =

is

is equal to

is the same as

the result is

gives

was

will be

The steps we use to translate a sentence into an equation are listed below.

Translate and Solve Applications

Most of the time a question that requires an algebraic solution comes out of a real life question. To begin with that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.

We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q for the number of quarters if you were solving a problem about coins.

Key Concepts

To Determine Whether a Number is a Solution to an Equation

Substitute the number in for the variable in the equation.

Simplify the expressions on both sides of the equation.

Determine whether the resulting statement is true.

If it is true, the number is a solution.

If it is not true, the number is not a solution.

Addition Property of Equality

For any numbers a, b, and c, if a=b, then a+c=b+c.

Subtraction Property of Equality

For any numbers a, b, and c, if a=b, then a−c=b−c.

To Translate a Sentence to an Equation

Locate the “equals” word(s). Translate to an equal sign (=).

Translate the words to the left of the “equals” word(s) into an algebraic expression.

Translate the words to the right of the “equals” word(s) into an algebraic expression.

To Solve an Application

Read the problem. Make sure all the words and ideas are understood.

Identify what we are looking for.

Name what we are looking for. Choose a variable to represent that quantity.

Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.

Solve the equation using good algebra techniques.

Check the answer in the problem and make sure it makes sense.

Answer the question with a complete sentence.

Glossary

solution of an equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.