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Solving Equations by Multiplying or Dividing

Objective Learn to solve multiplication and division equations.

This lesson uses one of the most important ideas in algebra, the Multiplication Property of Equality. Make sure you understand how it is being used.

 

Solving Equations by Multiplying or Dividing

Luisa is three times as old as Justin. If Luisa is 24 years old, how old is Justin?

In this problem, two quantities are equal to each other. In words, three times Justin's age is equal to Luisa's age. Luisa's age is 24, but we don't know Justin's age. When we do not know a quantity, we choose a letter as a placeholder for that value so we can work with the other numbers in the problem and perhaps determine the unknown quantity. The letter is called a variable. In this case, let's write j for Justin's age. The quantity "three times Justin's age" is now represented by 3j. Since three times Justin's age is equal to Luisa's age, we now write

3j = 24.

This expression is called an equation. If an equality results when a number is substituted for j, the number is a solution to the equation.

In order to determine j, we'll need to use an important property of equations.

Key Idea

Multiplying or dividing any equation by a nonzero number results in a true equation. Any solution of the original equation will be a solution of the new equation, and any solution of the new equation will be a solution of the original equation.

These ideas are called the Multiplication and Division Properties of Equality. In 3j = 24, we want to isolate the variable j on the left side, so that the equation reads

j = ?

where the ? is a quantity we have to determine that doesn't involve the variable j. Once the equation is in this form, it is solved since weknow that j is equal to the number on the right side.

Let' s solve the equation to dete rmine Justin's age.

3j = 24  
Divide each side by 3
j = 8 Simplify.

So Justin is 8 years old. Let's solve another equation using the Division Property of Equality.

 

Example 1

Solve 7x = 112.

Solution

7x = 112  
Divide each side by 7
x = 16 Simplify.

The solution is 16.

The next example shows how to solve an equation using the Multiplication Property of Equality.

 

Example 2

Solve .

Solution

 
Multiply each side by 12
x = 16 Simplify.

The solution is 48.

How does this work? To answer this question, think of the general pattern. For an equation of the form

(coefficient) · x = number, we divide the whole equation by the coefficient to get

.

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