FreeAlgebra Tutorials!   Home 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 Powers 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 http: 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 Fractions 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 Adding Quadratic Functions Conjugates Factoring 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 Percents Arithmetics 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 Polynomials Laws of Exponents Multiplying Polynomials Vertical Line Test http: Solving Inequalities with Fractions and Parentheses http: Multiplying Polynomials Fractions Solving Quadratic and Polynomial Equations Extraneous Solutions Fractions

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# 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) Â· x = number, we divide the whole equation by the coefficient to get .