Solve Equations with Fractions

4.7 Solve Equations with Fractions

The topics covered in this section are:

  1. Determine whether a fraction is a solution of an equation
  2. Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality
  3. Solve equations using the Multiplication Property of Equality
  4. Translate sentences to equations and solve

4.7.1 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. Substitute the number for the variable in the equation.
  2. Simplify the expressions on both sides of the equation.
  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 1

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. $x= – \frac{4}{5}$
Solution

Part 1.
$x – \frac{3}{10} = \frac{1}{2}$
Substitute $1$ for $x$.$1 – \frac{3}{10} \stackrel{?}{=} \frac{1}{2}$
Change to fractions with a LCD of $10$.$\frac{10}{10} – \frac{3}{10} \stackrel{?}{=} \frac{5}{10}$
Subtract.$\frac{7}{10} \neq \frac{5}{10}$

Since $x=1$ does not result in a true equation, $1$ is not a solution to the equation.

Part 2.
$x – \frac{3}{10} = \frac{1}{2}$
Substitute $\frac{4}{5}$ for $x$.$\frac{4}{5} – \frac{3}{10} \stackrel{?}{=} \frac{1}{2}$
$\frac{8}{10} – \frac{3}{10} \stackrel{?}{=} \frac{5}{10}$
Subtract.$\frac{5}{10} = \frac{5}{10}$ ✓

Since $x= \frac{4}{5}$ results in a true equation, $\frac{4}{5}$ is a solution to the equation $x – \frac{3}{10} = \frac{1}{2}$.

Part 3.
$x – \frac{3}{10} = \frac{1}{2}$
Substitute $- \frac{4}{5}$ for $x$.$- \frac{4}{5} – \frac{3}{10} \stackrel{?}{=} \frac{1}{2}$
$- \frac{8}{10} – \frac{3}{10} \stackrel{?}{=} \frac{5}{10}$
Subtract.$- \frac{11}{10} \neq \frac{5}{10}$

Since $x= – \frac{4}{5}$ does not results in a true equation, $- \frac{4}{5}$ is not a solution to the equation.

4.7.2 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 $\frac{a}{c} = \frac{b}{c}, c \neq 0$.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 2

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

Solution

$y+ \frac{9}{16} = \frac{5}{16}$
Subtract $\frac{9}{16}$ from each side to undo the addition.$y+ \frac{9}{16} – \frac{9}{16} = \frac{5}{16} – \frac{9}{16}$
Simplify on each side of the equation.$y+0=- \frac{4}{16}$
Simplify the fraction.$y=- \frac{1}{4}$
Check:$y+ \frac{9}{16} = \frac{5}{16}$
Substitute: $y=- \frac{1}{4}$$- \frac{1}{4}+ \frac{9}{16} \stackrel{?}{=} \frac{5}{16}$
Rewrite as fractions with the LCD.$- \frac{4}{16}+ \frac{9}{16} \stackrel{?}{=} \frac{5}{16}$
Add.$\frac{5}{16} = \frac{5}{16}$

Since $y=- \frac{1}{4}$ makes $y+ \frac{9}{16} = \frac{5}{16}$ a true statement, we know we have found the solution to this equation.

We used the Subtraction Property of Equality in Example 2. Now we’ll use the Addition Property of Equality.

Example 3

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

Solution

$a- \frac{5}{9} = – \frac{8}{9}$
Add $\frac{5}{9}$ from each side to undo the addition.$a- \frac{5}{9} + \frac{5}{9} = – \frac{8}{9} + \frac{5}{9}$
Simplify on each side of the equation.$a+0=- \frac{3}{9}$
Simplify the fraction.$a=- \frac{1}{3}$
Check:$a- \frac{5}{9} = – \frac{8}{9}$
Substitute: $a=- \frac{1}{3}$$- \frac{1}{3}- \frac{5}{9} \stackrel{?}{=} – \frac{8}{9}$
Change to common denominator.$- \frac{3}{9}- \frac{5}{9} \stackrel{?}{=} – \frac{8}{9}$
Subtract.$- \frac{8}{9} = – \frac{8}{9}$

Since $a= – \frac{1}{3}$ makes the equation true, we know that $a= – \frac{1}{3}$ is the solution to the equation.

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

Example 4

Solve: $10q = 44$.

Solution

$10q=44$
Divide both sides by $10$ to undo the multiplication.$\frac{10q}{10} = \frac{44}{10}$
Simplify.$q=\frac{22}{5}$
Check:$10( \frac{22}{5} ) \stackrel{?}{=} 44$
Substitute: $q= \frac{22}{5}$ into the original equation.$10( \frac{22}{5} ) \stackrel{?}{=} 44$
Simplify.$\stackrel{2}{\cancel{10}} ( \frac{22}{\cancel{5}} ) \stackrel{?}{=} 44$
Multiply.$44=44$

The solution to the equation was the fraction $\frac{22}{5}$. We leave it as an improper fraction.

4.7.3 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 $c$,

if $a=b$, 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 $\frac{x}{7} = -9$.

Example 5

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

Solution

$\frac{x}{7} = -9$
Use the Multiplication Property of Equality to multiply both sides by $7$. This will isolate the variable.$7 \cdot \frac{x}{7} = 7(-9)$
Multiply.$\frac{7x}{7} = -63$
Simplify.$x=-63$
Check:
Substitute $-63$ for $x$ in the original equation.
$\frac{-63}{7} \stackrel{?}{=} -9$
The equation is true.$-9=-9$

Example 6

Solve: $- \frac{9}{-8} = -40$.

Solution

Here, $p$ is divided by $-8$. We must multiply by $-8$ to isolate $p$.

$- \frac{9}{-8} = -40$
Multiply both sides by $-8$$-8( \frac{p}{-8} ) = -8(-40)$
Multiply.$\frac{-8p}{-8} = 320$
Simplify.$p=320$
Check:
Substitute $p=320$.$\frac{320}{-8} \stackrel{?}{=} -40$
The equation is true.$-40=-40$

Solve Equations with a Coefficient of $-1$

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 7.

Example 7

Solve: $-y=15$.

Solution

One way to solve the equation is to rewrite $-y$ as $-1y$, and then use the Division Property of Equality to isolate $y$.

$-y=15$
Rewrite $-y$ as $-1y$.$-1y=15$
Divide both sides by $-1$.$\frac{-1y}{-1} = \frac{15}{-1}$
Simplify each side.$y=-15$

Another way to solve this equation is to multiply both sides of the equation by $-1$.

$-y=15$
Multiply both sides by $-1$.$-1(-y)=(-1)15$
Simplify each side.$y=-15$

The third way to solve the equation is to read $-y$ as “the opposite of $y$.” What number has $15$ as its opposite? The opposite of $15$ is $-15$. So $y=-15$.

For all three methods, we isolated $y$ and solve the equation.

Check:

$-y=15$
Substitute $y=-15$.$-(-15) \stackrel{?}{=} (15)$
Simplify each side.$15=15$

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:

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

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 8.

Example 8

Solve: $\frac{3}{4} x = 24$.

Solution
$\frac{3}{4} x =24$
Multiply both sides by the reciprocal of the coefficient.$\frac{4}{3} \cdot \frac{3}{4} x = \frac{4}{3} \cdot 24$
Simplify.$1x = \frac{4}{3} \cdot \frac{24}{1}$
Multiply.$x=32$
Check:$\frac{3}{4} x =24$
Substitute $x=32$.$\frac{3}{4} \cdot 32 \stackrel{?}{=} 24$
Rewrite $32$ as a fraction.$\frac{3}{4} \cdot \frac{32}{1} \stackrel{?}{=} 24$
Multiply. The equation is true.$24=24$

Notice that in the equation $\frac{3}{4} x =24$, we could have divided both sides by $\frac{3}{4}$ to get $x$ 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.

Example 9

Solve: $- \frac{3}{8} w = 72$.

Solution

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.

$- \frac{3}{8} w =72$
Multiply both sides by the reciprocal of $- \frac{3}{8}$.$- \frac{8}{3} (- \frac{3}{8} w ) = (- \frac{8}{3} ) 72$
Simplify; reciprocals multiply to one.$1w = – \frac{8}{3} \cdot \frac{72}{1}$
Multiply.$w=-192$
Check:$- \frac{3}{8} w =72$
Let $w=-192$.$- \frac{3}{8} (-192) \stackrel{?}{=} 72$
Multiply. It checks.$72=72$

7.4.4 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 $a,b$, and $c$,
if $a=b$, then $a-c=b-c$.
Addition Property of Equality:
For any real numbers $a,b$, and $c$,
if $a=b$, then $a+c=b+c$.
Division Property of Equality:
For any real numbers $a,b$, and $c$ where $c \neq 0$,
if $a=b$, then $\frac{a}{c} = \frac{b}{c}$.
Multiplication Property of Equality:
For any real numbers $a,b$, and $c$,
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 10

Translate and solve: $n$ divided by $6$ is $-24$.

Solution

Translate..
Multiply both sides by $6$.$6 \cdot \frac{n}{6} = 6(-24)$
Simplify.$n=-144$
Check:Is $-144$ divided by $6$ equal to $-24$?
Translate.$\frac{-144}{6} \stackrel{?}{=} -24$
Simplify. It checks.$-24=-24$

Example 11

Translate and solve: The quotient of $q$ and $-5$ is $70$.

Solution

Translate..
Multiply both sides by $-5$.$-5( \frac{q}{-5} ) = -5(70)$
Simplify.$q=-350$
Check:Is the quotient of $-350$ and $-5$ equal to $70$?
Translate.$\frac{-350}{-5} \stackrel{?}{=} 70$
Simplify. It checks.$70=70$

Example 12

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

Solution

Translate..
Multiply both sides by $\frac{3}{2}$.$\frac{3}{2} \cdot \frac{2}{3} f = \frac{3}{2} \cdot 18$
Simplify.$f=27$
Check:Is two-thirds of $27$ equal to $18$?
Translate.$\frac{2}{3} (27) \stackrel{?}{=} 18$
Simplify. It checks.$18=18$

Example 13

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

Solution

The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4}$.
Translate.$\frac{m}{\frac{5}{6}} = \frac{3}{4}$
Multiply both sides by $\frac{5}{6}$ to isolate $m$.$\frac{5}{6} ( \frac{m}{\frac{5}{6}} ) = \frac{5}{6} ( \frac{3}{4} )$
Simplify.$m= \frac{5 \cdot 3}{6 \cdot 4}$
Remove common factors and multiply.$m = \frac{5}{8}$
Check:
The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4}$.$\frac{\frac{5}{8}}{\frac{5}{6}} \stackrel{?}{=} \frac{3}{4}$
Rewrite as division.$\frac{5}{8} \div \frac{5}{6} \stackrel{?}{=} \frac{3}{4}$
Multiply the first fraction by the reciprocal of the second.$\frac{5}{8} \cdot \frac{6}{5} \stackrel{?}{=} \frac{3}{4}$
Simplify.$\frac{3}{4}=\frac{3}{4}$

Our solution checks.

Example 14

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

Solution

Translate..
Use the Subtraction Property of Equality to subtract $\frac{3}{8}$ from both sides.$\frac{3}{8} + x – \frac{3}{8} = 3 \frac{1}{2} – \frac{3}{8}$
Combine like terms on the left side.$x=3 \frac{1}{2} – \frac{3}{8}$
Convert mixed number to improper fraction.$x= \frac{7}{2} – \frac{3}{8}$
Convert to equivalent fractions with LCD of $8$.$x= \frac{28}{8} – \frac{3}{8}$
Subtract.$x= \frac{25}{8}$
Write as a mixed number.$x= 3 \frac{1}{8}$

We write the answer as a mixed number because the original problem used a mixed number.

Check:

Is the sum of three-eighths and $3 \frac{1}{8}$ equal to three and one-half?

$\frac{3}{8} + 3 \frac{1}{8} \stackrel{?}{=} 3 \frac{1}{2}$
Add.$3 \frac{4}{8} \stackrel{?}{=} 3 \frac{1}{2}$
Simplify.$3 \frac{1}{2} = 3 \frac{1}{2} $

The solution checks.

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