Solving Two-Step Linear Equations with Fractions

A linear equation is an equation in which the variable(s) are multiplied by or added to numbers with nothing more complicated than that. Linear equations don't contain square roots, exponents, $\frac{1}{x}$ , or other more complex components like that.

The solution for an equation is the number that can be plugged in for the variable to make a true math statement. For example, in the above statement $4x+9=21$ , if you plug 3 in for the x, you get $4\times 3+9=21$ , which is a true statement. $4\times 3=12$ , $12+9=21$ . Sometimes linear equations can have more than one correct solution.

One-step linear equations are generally in the form of $x+y=z$ or $x\times y=z$ . Two-step linear equations are in the form of something like $ax+y=z$ . In this tutorial, we're looking at two-step equations that have fractions in them.

Solving two-step linear equations

To solve a two-step linear equation, we need to use inverse operations to undo each operation in the equation in reverse PEMDAS order.

Example 1

Solve $5x+8=28$ .

Because we're going in reverse order, we are going to undo the addition first. So we subtract 8 from each side of the equation.

$5x+8-8=28-8$

Simplify by performing the subtraction.

$5x=20$

Next, we are going to undo the multiplication by using division. We will divide each side of the equation by 5, which will isolate x.

$\frac{5x}{5}=\frac{20}{5}$

Simplify by performing the division.

$x=4$

Finally, we check our answer by seeing if it works in the original equation.

$\left(5\times 4\right)+8=28$

$20+8=28$

It works, so we have found the solution for the variable x in the two-step linear equation.

Two-step linear equations with fractions

Solving a two-step linear equation with fractions involves undoing division instead of undoing multiplication. We still undo each operation in reverse order. Let's look at an example to see how to do this.

Example 2

$\frac{x}{5}-10=3$

Our first step is to undo the subtraction, so we will add 10 to each side.

$\frac{x}{5}-10+10=3+10$

Simplify by performing the addition.

$\frac{x}{5}=13$

Next, we undo the division by multiplying each side of the equation by 5.

$\frac{x}{5}\times 5=13\times 5$

Simplify by performing the multiplication.

$x=65$

Now let's check to see if our solution works in the original equation.

$\frac{65}{5}-10=3$

$13-10=3$

$3=3$

It does work, so we have found the correct solution. And that is how we work two-step linear equations with fractions.

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

Solving Systems of Linear Equations

Comparing Quantities

Factoring: b Negative

Flashcards covering the Solving Two-Step Linear Equations with Fractions

Algebra 1 Flashcards

College Algebra Flashcards

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

Algebra 1 Diagnostic Tests

College Algebra Diagnostic Tests

Get help learning to solve two-step linear equations with fractions

Tutoring is probably the best way to supplement in-class teaching on working with linear equations, including two-step linear equations with fractions. There are many steps to consider when working with multi-step linear equations, and some students need extra practice to master the process. A tutor will work at your student's pace and answer their questions as they arise, making tutoring an efficient and effective way for your student to study and learn two-step linear equations with fractions. A tutor can also cater to your student's learning style in their 1-on-1 sessions and can even provide customized learning materials to help your student thrive. At Varsity Tutors, we would be proud to set your student up with a tutor who is an excellent match to their academic needs. Simply contact us and speak with one of our friendly Educational Directors to get started.