Find All Real Solutions of the Equation With Fractions

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

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 $x+4$ when $x=\mathrm{-3}$
If you missed this problem, review Example 3.23.

Be Prepared 4.18

Solve: $2y-3=9.$
If you missed this problem, review Example 3.61.

Be Prepared 4.19

Solve: $y-3=\mathrm{-9}$
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.

1. Step 1. Substitute the number for the variable in the equation.
2. Step 2. Simplify the expressions on both sides of the equation.
3. 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 $x-\frac{3}{10}=\frac{1}{2}.$

1. ⓐ $x=1$
2. ⓑ $x=\frac{4}{5}$
3. ⓒ $x=-\frac{4}{5}$

Try It 4.189

Determine whether each number is a solution of the given equation.

$x-\frac{2}{3}=\frac{1}{6}$ :

1. ⓐ $x=1$
2. ⓑ $x=\frac{5}{6}$
3. ⓒ $x=-\frac{5}{6}$

Try It 4.190

Determine whether each number is a solution of the given equation.

$y-\frac{1}{4}=\frac{3}{8}$ :

1. ⓐ $y=1$
2. ⓑ $y=-\frac{5}{8}$
3. ⓒ $y=\frac{5}{8}$

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 $a,b,$ and $c,$

if $a=b,$ then $a+c=b+c.$	Addition Property of Equality
$$ if $a=b,$ then $a-c=b-c.$	Subtraction Property of Equality
if $a=b,$ then $\frac{a}{c}=\frac{b}{c},c\ne 0.$	Division Property of Equality

Table 4.3

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: $y+\frac{9}{16}=\frac{5}{16}.$

Try It 4.191

Solve: $y+\frac{11}{12}=\frac{5}{12}.$

Try It 4.192

Solve: $y+\frac{8}{15}=\frac{4}{15}.$

We used the Subtraction Property of Equality in Example 4.96. Now we'll use the Addition Property of Equality.

Example 4.97

Solve: $a-\frac{5}{9}=-\frac{8}{9}.$

Try It 4.193

Solve: $a-\frac{3}{5}=-\frac{8}{5}.$

Try It 4.194

Solve: $n-\frac{3}{7}=-\frac{9}{7}.$

The next example may not seem to have a fraction, but let's see what happens when we solve it.

Example 4.98

Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation $\frac{x}{4}=3.$ We want to know what number divided by $4$ gives $3.$ So to "undo" the division, we will need to multiply by $4.$ 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 $a,b,$ and $c,$

$$\text{if}a=b,\text{then}ac=bc.$$

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 $\frac{x}{7}=\mathrm{-9}.$

Example 4.99

Example 4.100

Try It 4.200

Solve: $\frac{x}{\mathrm{-11}}=\mathrm{-12}.$

Solve Equations with a Coefficient of $\mathrm{-1}$

Look at the equation $-y=15.$ Does it look as if $y$ is already isolated? But there is a negative sign in front of $y,$ 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

Try It 4.201

Try It 4.202

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 $1.$

For example, in the equation:

$$\frac{3}{4}x=24$$

The coefficient of $x$ is $\frac{3}{4}.$ To solve for $x,$ we need its coefficient to be $1.$ Since the product of a number and its reciprocal is $1,$ our strategy here will be to isolate $x$ by multiplying by the reciprocal of $\frac{3}{4}.$ We will do this in Example 4.102.

Example 4.102

Try It 4.203

Try It 4.204

Example 4.103

Try It 4.205

Try It 4.206

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 $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$ if $a=b,$ then $a-c=b-c.$	Addition Property of Equality: For any real numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$ if $a=b,$ then $a+c=b+c.$
Division Property of Equality: For any numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$ where $\mathit{\text{c}}\ne \mathit{0}$ if $a=b,$ then $\frac{a}{c}=\frac{b}{c}$	Multiplication Property of Equality: For any real numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c}}$ if $a=b,$ then $ac=bc$

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: $n$ divided by $6$ is $\mathrm{-24}.$

Try It 4.207

Translate and solve: $n$ divided by $7$ is equal to $\mathrm{-21}.$

Try It 4.208

Translate and solve: $n$ divided by $8$ is equal to $\mathrm{-56}.$

Example 4.105

Translate and solve: The quotient of $q$ and $\mathrm{-5}$ is $70.$

Try It 4.209

Translate and solve: The quotient of $q$ and $\mathrm{-8}$ is $72.$

Try It 4.210

Translate and solve: The quotient of $p$ and $\mathrm{-9}$ is $81.$

Example 4.106

Translate and solve: Two-thirds of $f$ is $18.$

Try It 4.211

Translate and solve: Two-fifths of $f$ is $16.$

Try It 4.212

Translate and solve: Three-fourths of $f$ is $21.$

Example 4.107

Translate and solve: The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4}.$

Try It 4.213

Translate and solve. The quotient of $n$ and $\frac{2}{3}$ is $\frac{5}{12}.$

Try It 4.214

Translate and solve The quotient of $c$ and $\frac{3}{8}$ is $\frac{4}{9}.$

Example 4.108

Translate and solve: The sum of three-eighths and $x$ is three and one-half.

Try It 4.215

Translate and solve: The sum of five-eighths and $x$ is one-fourth.

Try It 4.216

Translate and solve: The difference of one-and-three-fourths and $x$ is five-sixths.

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.

498 .

$x-\frac{2}{5}=\frac{1}{10}$ :

1. ⓐ $x=1$
2. ⓑ $x=\frac{1}{2}$
3. ⓒ $x=-\frac{1}{2}$

499.

$y-\frac{1}{3}=\frac{5}{12}$ :

1. ⓐ $y=1$
2. ⓑ $y=\frac{3}{4}$
3. ⓒ $y=-\frac{3}{4}$

500 .

$h+\frac{3}{4}=\frac{2}{5}$ :

1. ⓐ $h=1$
2. ⓑ $h=\frac{7}{20}$
3. ⓒ $h=-\frac{7}{20}$

501.

$k+\frac{2}{5}=\frac{5}{6}$ :

1. ⓐ $k=1$
2. ⓑ $k=\frac{13}{30}$
3. ⓒ $k=-\frac{13}{30}$

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

508 .

$x-\left(-\frac{3}{20}\right)=-\frac{11}{20}$

509.

$z-\left(-\frac{5}{12}\right)=-\frac{7}{12}$

512 .

$s+\left(-\frac{1}{2}\right)=-\frac{8}{9}$

513.

$k+\left(-\frac{1}{3}\right)=-\frac{4}{5}$

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.

550 .

$n$ divided by eight is $\mathrm{-16}.$

551.

$n$ divided by six is $\mathrm{-24}.$

552 .

$m$ divided by $\mathrm{-9}$ is $\mathrm{-7}.$

553.

$m$ divided by $\mathrm{-7}$ is $\mathrm{-8}.$

554 .

The quotient of $f$ and $\mathrm{-3}$ is $\mathrm{-18}.$

555.

The quotient of $f$ and $\mathrm{-4}$ is $\mathrm{-20}.$

556 .

The quotient of $g$ and twelve is $8.$

557.

The quotient of $g$ and nine is $14.$

558 .

Three-fourths of $q$ is $12.$

559.

Two-fifths of $q$ is $20.$

560 .

Seven-tenths of $p$ is $\mathrm{-63}.$

561.

Four-ninths of $p$ is $\mathrm{-28}.$

562 .

$m$ divided by $4$ equals negative $6.$

563.

The quotient of $h$ and $2$ is $43.$

564 .

Three-fourths of $z$ is $15.$

565.

The quotient of $a$ and $\frac{2}{3}$ is $\frac{3}{4}.$

566 .

The sum of five-sixths and $x$ is $\frac{1}{2}.$

567.

The sum of three-fourths and $x$ is $\frac{1}{8}.$

568 .

The difference of $y$ and one-fourth is $-\frac{1}{8}.$

569.

The difference of $y$ and one-third is $-\frac{1}{6}.$

Everyday Math

570 .

Shopping Teresa bought a pair of shoes on sale for $\$48$ . The sale price was $\frac{2}{3}$ of the regular price. Find the regular price of the shoes by solving the equation $\frac{2}{3}p=48$

571.

Playhouse The table in a child's playhouse is $\frac{3}{5}$ of an adult-size table. The playhouse table is $18$ inches high. Find the height of an adult-size table by solving the equation $\frac{3}{5}h=18.$

Writing Exercises

572 .

Example 4.100 describes three methods to solve the equation $-y=15.$ Which method do you prefer? Why?

573.

Richard thinks the solution to the equation $\frac{3}{4}x=24$ is $16.$ 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?

Find All Real Solutions of the Equation With Fractions

Source: https://openstax.org/books/prealgebra-2e/pages/4-7-solve-equations-with-fractions

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