Learning Objectives
By the end of this section, you will be able to:
- Determine whether a fraction is a solution of an equation
- Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality
- Solve equations using the Multiplication Property of Equality
- Translate sentences to equations and solve
Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.
Evaluate
If you missed this problem, review Example 3.23.
Solve:
If you missed this problem, review Example 3.61.
Solve:
If you missed this problem, review Example 4.28.
Determine Whether a Fraction is a Solution of an Equation
As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.
The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction.
How To
Determine whether a number is a solution to an equation.
- Step 1. Substitute the number for the variable in the equation.
- Step 2. Simplify the expressions on both sides of the equation.
- Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.
Example 4.95
Determine whether each of the following is a solution of
- ⓐ
- ⓑ
- ⓒ
Determine whether each number is a solution of the given equation.
- ⓐ
- ⓑ
- ⓒ
Determine whether each number is a solution of the given equation.
- ⓐ
- ⓑ
- ⓒ
Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality
In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.
Addition, Subtraction, and Division Properties of Equality
For any numbers
Addition Property of Equality | |
Subtraction Property of Equality | |
Division Property of Equality |
In other words, when you add or subtract the same quantity from both sides of an equation, or divide both sides by the same quantity, you still have equality.
Example 4.96
Solve:
Solve:
Solve:
We used the Subtraction Property of Equality in Example 4.96. Now we’ll use the Addition Property of Equality.
Example 4.97
Solve:
Solve:
Solve:
The next example may not seem to have a fraction, but let’s see what happens when we solve it.
Example 4.98
Solve:
Solve:
Solve:
Solve Equations with Fractions Using the Multiplication Property of Equality
Consider the equation We want to know what number divided by gives So to “undo” the division, we will need to multiply by 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.
The Multiplication Property of Equality
For any numbers and
If you multiply both sides of an equation by the same quantity, you still have equality.
Let’s use the Multiplication Property of Equality to solve the equation
Example 4.99
Solve:
Solve:
Solve:
Example 4.100
Solve:
Solve:
Solve:
Solve Equations with a Coefficient of
Look at the equation Does it look as if is already isolated? But there is a negative sign in front of so it is not isolated.
There are three different ways to isolate the variable in this type of equation. We will show all three ways in Example 4.101.
Example 4.101
Solve:
Try It 4.201
Solve:
Try It 4.202
Solve:
Solve Equations with a Fraction Coefficient
When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to
For example, in the equation:
The coefficient of is To solve for we need its coefficient to be Since the product of a number and its reciprocal is our strategy here will be to isolate by multiplying by the reciprocal of We will do this in Example 4.102.
Example 4.102
Solve:
Try It 4.203
Solve:
Try It 4.204
Solve:
Example 4.103
Solve:
Try It 4.205
Solve:
Try It 4.206
Solve:
Translate Sentences to Equations and Solve
Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.
Subtraction Property of Equality: For any real numbers and if then |
Addition Property of Equality: For any real numbers and if then |
Division Property of Equality: For any numbers and where if then |
Multiplication Property of Equality: For any real numbers and if then |
When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.
In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions.
Example 4.104
Translate and solve: divided by is
Translate and solve: divided by is equal to
Translate and solve: divided by is equal to
Example 4.105
Translate and solve: The quotient of and is
Translate and solve: The quotient of and is
Translate and solve: The quotient of and is
Example 4.106
Translate and solve: Two-thirds of is
Translate and solve: Two-fifths of is
Translate and solve: Three-fourths of is
Example 4.107
Translate and solve: The quotient of and is
Translate and solve. The quotient of and is
Translate and solve The quotient of and is
Example 4.108
Translate and solve: The sum of three-eighths and is three and one-half.
Translate and solve: The sum of five-eighths and is one-fourth.
Translate and solve: The difference of one-and-three-fourths and is five-sixths.
Media Access Additional Online Resources
Section 4.7 Exercises
Practice Makes Perfect
Determine Whether a Fraction is a Solution of an Equation
In the following exercises, determine whether each number is a solution of the given equation.
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- ⓑ
- ⓒ
- ⓐ
- ⓑ
- ⓒ
Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality
In the following exercises, solve.
Solve Equations with Fractions Using the Multiplication Property of Equality
In the following exercises, solve.
Mixed Practice
In the following exercises, solve.
Translate Sentences to Equations and Solve
In the following exercises, translate to an algebraic equation and solve.
divided by eight is
divided by is
The quotient of and is
The quotient of and twelve is
Three-fourths of is
Seven-tenths of is
divided by equals negative
Three-fourths of is
The sum of five-sixths and is
The difference of and one-fourth is
Everyday Math
Shopping Teresa bought a pair of shoes on sale for The sale price was of the regular price. Find the regular price of the shoes by solving the equation
Playhouse The table in a child’s playhouse is of an adult-size table. The playhouse table is inches high. Find the height of an adult-size table by solving the equation
Writing Exercises
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?