4.8: Solve Equations with Fractions
- Page ID
- 114904
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- 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
Be Prepared 4.17
Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.
Evaluate when
If you missed this problem, review Example 3.23.
Be Prepared 4.18
Solve:
If you missed this problem, review Example 3.61.
Be Prepared 4.19
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
- ⓐ
- ⓑ
- ⓒ
- Answer
ⓐ Change to fractions with a LCD of 10. Subtract. Since does not result in a true equation, is not a solution to the equation.
ⓑ Subtract.
Since results in a true equation, is a solution to the equation
ⓒ | |
Subtract. |
Since does not result in a true equation, is not a solution to the equation.
Try It 4.189
Determine whether each number is a solution of the given equation.
:
- ⓐ
- ⓑ
- ⓒ
Try It 4.190
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 and
if then | Addition Property of Equality |
if then | Subtraction Property of Equality |
if then | 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:
- Answer
Subtract from each side to undo the addition. Simplify on each side of the equation. Simplify the fraction. Check: Substitute . Rewrite as fractions with the LCD. Add. Since makes a true statement, we know we have found the solution to this equation.
Try It 4.191
Solve:
Try It 4.192
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:
- Answer
Add from each side to undo the subtraction. Simplify on each side of the equation. Simplify the fraction. Check: Substitute . Change to common denominator. Subtract. Since makes the equation true, we know that is the solution to the equation.
Try It 4.193
Solve:
Try It 4.194
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:
- Answer
Divide both sides by 10 to undo the multiplication. Simplify. Check: Substitute into the original equation. Simplify. Multiply. The solution to the equation was the fraction We leave it as an improper fraction.
Try It 4.195
Solve:
Try It 4.196
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:
- Answer
Use the Multiplication Property of Equality to multiply both sides by . This will isolate the variable. Multiply. Simplify. The equation is true.
Try It 4.197
Solve:
Try It 4.198
Solve:
Example 4.100
Solve:
- Answer
Here, is divided by We must multiply by to isolate
Multiply both sides by Multiply. Simplify. Check: Substitute . The equation is true.
Try It 4.199
Solve:
Try It 4.200
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:
- Answer
One way to solve the equation is to rewrite as and then use the Division Property of Equality to isolate
Rewrite as . Divide both sides by −1. Simplify each side. Another way to solve this equation is to multiply both sides of the equation by
Multiply both sides by −1. Simplify each side.
The third way to solve the equation is to read as “the opposite of .” What number has as its opposite? The opposite of is So
For all three methods, we isolated is isolated and solved the equation.
Check:
Substitute . | |
Simplify. The equation is true. |
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
Example 4.102
Solve:
- Answer
Multiply both sides by the reciprocal of the coefficient. Simplify. Multiply. Check: Substitute . Rewrite as a fraction. Multiply. The equation is true. Notice that in the equation we could have divided both sides by to get by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.
Try It 4.203
Solve:
Try It 4.204
Solve:
Example 4.103
Solve:
- Answer
The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.
Multiply both sides by the reciprocal of . Simplify; reciprocals multiply to one. Multiply. Check: Let . Multiply. It checks.
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
- Answer
Translate. Multiply both sides by . Simplify. Check: Is divided by equal to ? Translate. Simplify. It checks.
Try It 4.207
Translate and solve: divided by is equal to
Try It 4.208
Translate and solve: divided by is equal to
Example 4.105
Translate and solve: The quotient of and is
- Answer
Translate. Multiply both sides by . Simplify. Check: Is the quotient of and equal to ? Translate. Simplify. It checks.
Try It 4.209
Translate and solve: The quotient of and is
Try It 4.210
Translate and solve: The quotient of and is
Example 4.106
Translate and solve: Two-thirds of is
- Answer
Translate. Multiply both sides by . Simplify. Check: Is two-thirds of equal to ? Translate. Simplify. It checks.
Try It 4.211
Translate and solve: Two-fifths of is
Try It 4.212
Translate and solve: Three-fourths of is
Example 4.107
Translate and solve: The quotient of and is
- Answer
The quotient of and is . Translate. Multiply both sides by to isolate . Simplify. Remove common factors and multiply. Check: Is the quotient of and equal to ? Rewrite as division. Multiply the first fraction by the reciprocal of the second. Simplify. Our solution checks.
Try It 4.213
Translate and solve. The quotient of and is
Try It 4.214
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.
- Answer
Translate. Use the Subtraction Property of Equality to subtract from both sides. Combine like terms on the left side. Convert mixed number to improper fraction. Convert to equivalent fractions with LCD of 8. Subtract. Write as a mixed number. We write the answer as a mixed number because the original problem used a mixed number.
Check:
Is the sum of three-eighths and equal to three and one-half?
Add. Simplify.
The solution checks.
Try It 4.215
Translate and solve: The sum of five-eighths and is one-fourth.
Try It 4.216
Translate and solve: The difference of one-and-three-fourths and is five-sixths.
Media
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.
:
- ⓐ
- ⓑ
- ⓒ
:
- ⓐ
- ⓑ
- ⓒ
:
- ⓐ
- ⓑ
- ⓒ
:
- ⓐ
- ⓑ
- ⓒ
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 six is
divided by is
divided by is
The quotient of and is
The quotient of and is
The quotient of and twelve is
The quotient of and nine is
Three-fourths of is
Two-fifths of is
Seven-tenths of is
Four-ninths of is
divided by equals negative
The quotient of and is
Three-fourths of is
The quotient of and is
The sum of five-sixths and is
The sum of three-fourths and is
The difference of and one-fourth is
The difference of and one-third 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
Example 4.100 describes three methods to solve the equation Which method do you prefer? Why?
Richard thinks the solution to the equation is Explain why Richard is wrong.
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?