Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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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.


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 .


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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