In earlier lessons, you learned to solve one-step equations involving integers, decimals, and fractions. Now you will learn to solve two-step equations. In these types of equations, it is better to perform addition and subtraction before multiplication and division. In other words, you still use inverse operations to solve these types of equations, but for these equations, you will reverse the order of operations when solving them. This rule remains in effect for any equation involving more than one operation. The ultimate goal in solving a two-step equation is the same as the goal of solving a one-step equation: to isolate the variable on one side of the equation.
Let's get started!
Example 1
Solve two-step equations using division:
2x+4=12\\
-4\:\:\:-4 \:\textcolor{green}{\text{Subtract 4 from both sides.}}\\
2x=8\\
\frac{2x}{2}=\frac{8}{2}\:\textcolor{green}{\text{Divide both sides by 2.}}\\
x=4
Check your solution:-2\left(4\right)+4=12\\
8+4=12
20=-2y-6\\
+6\:\:\:+6 \:\textcolor{green}{\text{Add 6 to both sides.}}\\
26=-2y\\
\frac{26}{-2}=\frac{-2y}{-2}\:\textcolor{green}{\text{Divide both sides by -2.}}\\
-13=y
Check your solution:20=-2\left(-13\right)-6\\
20=26-6\\
20=20
Example 2
Solve two-step equations by multiplying:
5+\frac{n}{6}=11\\
\textcolor{green}{-5\hspace{.75cm} -5\:\: \text{ Subtract 5 from both sides.}}\\
\frac{n}{6}=6\\
\textcolor{green}{6}\cdot \frac{n}{6}=6\cdot \textcolor{green}{6 \:\: \text{Multiply both sides by 6}}\\
n=36
Check your solution:5+\frac{36}{6}=11;\:5+6=11
8=\frac{z}{8}-3 \\
\textcolor{green}{+3\hspace{.75cm} +3 \:\:\text{Add 3 to both sides.}}\\
11=\frac{z}{8}\\
\textcolor{green}{8}\cdot 11=\frac{z}{8}\cdot \textcolor{green}{8 \text{ Multiply both sides by 8}}\\
88=z
Check your solution: 8=\frac{88}{8-3};\:8=11-3;\:8=8
Investigate
A new one-year membership at Silver Fitness Center costs $300. A registration fee of $30 is paid up front and the membership cost is paid monthly. How much do new members pay each month?
Strategy
Summarize the problem, using only key words or phrases.
The registration fee is a onetime payment and the $300 will be divided over 12 months.
Registration fee plus 1 year - monthly cost is $300(Let m represent the monthly cost)
So $30 + m = $300.00
30+12m=300 \\
\textcolor{green}{-30 \hspace{.75cm} -30 \:\: \text{Subtract 30 from both sides}}\\
12m=270\\
\frac{12m}{12}=\frac{270}{12} \textcolor{green}{\text{ Divide both sides by 12}}\\
m=22.5
Check your solution:30+12(22.5)=300;\:30+270=300
Interpret the solution: It will cost $22.50 per month for a one year membership.
CHECK THIS OUT!Did you know that you can rewrite decimal equations and fraction equations as whole number equations simply by using the Distributive Property? This strategy makes solving these problems so much simpler! Look at the following examples to see if this is a strategy that you might want to use when solving two-step fraction or decimal problems.
Example 1: Writing Equivalent Equations without Fractions.
Write an equivalent equation for\frac{1}{3}x-4=\frac{2}{5}that does not contain fractions. Then solve that equation.
\frac{1}{3}x-4=\frac{2}{5} \textcolor{green}{\text{ The LCM of the denominators is 15.}}\\
\textcolor{green}{15}\cdot \frac{1}{3}x-\textcolor{green}{15}\cdot 4=\frac{2}{5}\cdot \textcolor{green}{15 \text{Multiply both sides by 15}}\\
\frac{15}{3}x-60=\frac{30}{5}\textcolor{green}{\text{Simplify}}\\
5x-60=6\textcolor{green}{\text{This is an equivalent equation}}\\
\hspace{.5cm}\textcolor{green}{+60\:\: +60 \:\:\text{Add 60 to both sides}}\\
\frac{5x}{5}=\frac{66}{5}\textcolor{green}{\:\:\text{Divide both sides by 5}}\\
x=13 \frac{1}{5}
Example 2: Rewrite an equivalent equations without decimals.
15.x+0.75=14.25 \textcolor{green}{\:\:\text{The equation has decimals to the hundredths, so multiply both sides by 100.}}\\
\textcolor{green}{100}(15.x+0.75)=\textcolor{green}{100}(14.25)\textcolor{green}{\:\text{Use the Distributive Property}}\\
150x+75=1425 \textcolor{green}{\:\text{This is an equivalent equation without decimals.}}\\
\:\:\:-75\:\:\:-75 \textcolor{green}{\:\text{Subtract 75 from both sides.}}\\
\frac{150x}{150}=\frac{1350}{150}\:\textcolor{green}{\text{Divide both sides by 150.}}\\
x=9
You can use number properties to help solve any two-step equation with rational numbers!
Now, take a look at this video to see more examples of solving two-step equations in different formats.
It is time for you to complete a bit of practice before you work on assignments.
Solve Two-Step Equations
Which equation in each pair has the larger solution?
1. a. 3=4+\frac{n}{15} or b.7+\frac{k}{16}=6
Solution: a
2. a. -2m + 1 = -3 or b. -3\frac{3}{10}+\frac{1}{4}n=\frac{73}{10}
Solve Two-Step Equations