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# Solving One-Step Linear Equations with Fractions

Some linear equations can be solved with just a single operation. For this type of equation, we use the inverse operation to solve it. As we recall, using the inverse operation is a process where you use subtraction to solve an addition problem, addition to solve a subtraction problem, multiplication to solve a division problem, or division to solve a multiplication problem.

For example, $6+4=10$ , so $10-6=4$ and $10-4=6$ .

If you had an equation to solve, $x+4=10$ , you could solve it by subtracting 4 from each side.

$x+4-4=10-4$

$x=6$

The same steps are used when solving linear equations with fractions.

## One-step linear equations with fractions: addition or subtraction

Example 1

Solve the following linear equation using inverse operations.

$a+\frac{1}{7}=\frac{3}{7}$

Since the inverse operation of addition is subtraction, subtract $\frac{1}{7}$ from each side.

$a+\frac{1}{7}-\frac{1}{7}=\frac{3}{7}-\frac{1}{7}$

Simplify.

$a=\frac{2}{7}$

Example 2

Solve the following linear equation using inverse operations.

$a-\frac{1}{6}=\frac{5}{12}$

For this problem, we first need to make the denominators match. Since 12 is the LCM of 6 and 12, the problem becomes:

$a-\frac{2}{12}=\frac{5}{12}$

Now use the inverse operation of addition to add $\frac{2}{12}$ to each side of the equation.

$a-\frac{2}{12}+\frac{2}{12}=\frac{5}{12}+\frac{2}{12}$

Simplify.

$a=\frac{7}{12}$

## One-step linear equations with fractions: multiplication or division

When solving multiplication or division linear equations with fractions using multiplication or division, if there is a coefficient in front of the variable, multiply by the reciprocal of that number to get a coefficient of 1. Recall that a reciprocal is an opposite fraction, such as 3 and $\frac{1}{3}$ (because 3 is $\frac{3}{1}$ ) or $\frac{5}{9}$ and $\frac{9}{5}$ . Multiplying reciprocals always gives a value of 1.

Example 3

Solve the following equation using inverse operations.

$\frac{y}{4}=300$

The inverse operation of division is multiplication, so multiply each side by 4.

$\frac{y}{4}×4=300×4$

Simplify

$y=1200$

Example 4

Solve the following equation using inverse operations.

$26=-\frac{13}{6}x$

To isolate the variable x (to get a coefficient of 1), multiply both sides by $-\frac{6}{13}$ , which is the reciprocal of $-\frac{13}{6}$ .

$26×\left(-\frac{6}{13}\right)=-\frac{13}{6}x×\left(-\frac{6}{13}\right)$

Simplify.

$x=-12$

## Practice questions on solving one-step linear equations with fractions

a. Solve the following linear equation using inverse operations.

$a+\frac{3}{15}=\frac{11}{15}$

Since subtraction is the inverse operation of addition, subtract 3/15 from each side.

$a+\frac{3}{15}-\frac{3}{15}=\frac{11}{15}-\frac{3}{15}$

Simplify.

$a=\frac{8}{15}$

b. Solve the following equation using inverse operations.

$a-\frac{3}{5}=\frac{1}{3}$

The first step is to make the denominators the same. Since the LCM of 5 and 3 is 15, the problem can be rewritten as:

$a-\frac{9}{15}=\frac{5}{15}$

Now, because addition is the inverse operation of subtraction, add 9/15 to each side of the equation.

$a-\frac{9}{15}+\frac{9}{15}=\frac{5}{15}+\frac{9}{15}$

Simplify.

$a=\frac{14}{15}$

c. Solve the following equation using inverse operations.

$\frac{y}{3}=200$

Since the inverse operation of division is multiplication, multiply each side by 3.

$\frac{y}{3}×3=200×3$

Simplify.

$y=600$

## Get help learning about solving one-step linear equations with fractions

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