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.

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

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 $- \frac{13}{6}$ .

$26 \times (- \frac{6}{13}) = - \frac{13}{6} x \times (- \frac{6}{13})$

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.

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

Simplify.

$y = 600$

Topics related to the Solving One-Step Linear Equations with Fractions

Ordered Pair

Dividing Fractions

Solving One-Step Linear Equations: Division

Flashcards covering the Solving One-Step Linear Equations with Fractions

6th Grade Math Flashcards

Common Core: 6th Grade Math Flashcards

Practice tests covering the Solving One-Step Linear Equations with Fractions

MAP 6th Grade Math Practice Tests

6th Grade Math Practice Tests

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

Dealing with fractions can be tricky for some students, and working with them inside of linear equations can be even more problematic. If your student is having a hard time mastering one-step linear equations with fractions, connect them with a professional math tutor who can guide them through the steps of the problems until they understand the concepts thoroughly.

A tutor will work 1-on-1 with your student in a setting without distractions so your student can focus on the work being done. With the undivided attention of an expert, your student can work at their pace to learn how to solve one-step linear equations with fractions. When your student needs to take extra time, their tutor can slow down and walk them through each step of the problems. As your student comes to understand the topic better, their tutor can speed it up and move through the material more quickly. Your student's needs drive the pace of each lesson.

If you want to learn more about how math tutoring can help your student tackle one-step linear equations with fractions, contact the Educational Directors at Varsity Tutors today.