Multi-Step Equations
Andrew Gloag
Eve Rawley
Anne Gloag
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both
in the U.S. and worldwide. Using an open-content, web-based
collaborative model termed the FlexBook®, CK-12 intends to
pioneer the generation and distribution of high-quality educational
content that will serve both as core text as well as provide an
adaptive environment for learning, powered through the FlexBook
Platform®.
Copyright © 2013 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/terms.
Printed: October 25, 2013
AUTHORS
Andrew Gloag
Eve Rawley
Anne Gloag
www.ck12.org
C ONCEPT
Concept 1. Multi-Step Equations
1
Multi-Step Equations
Here you’ll learn how to solve equations that take several steps to isolate the unknown variable.
What if you had an equation like 3(x − 2) − x = 6 that required more than two steps to solve for the variable? After
completing this Concept, you’ll be able to solve multi-step equations like this one that require using the distributive
property and combining like terms.
Watch This
MEDIA
Click image to the left for more content.
CK-12 Foundation: 0307S Multi-Step Equations
Guidance
We’ve seen that when we solve for an unknown variable, it can take just one or two steps to get the terms in the right
places. Now we’ll look at solving equations that take several steps to isolate the unknown variable. Such equations
are referred to as multi-step equations.
In this section, we’ll simply be combining the steps we already know how to do. Our goal is to end up with all the
constants on one side of the equation and all the variables on the other side. We’ll do this by collecting like terms.
Don’t forget, like terms have the same combination of variables in them.
Example A
Solve
3x+4
3
− 5x = 6.
Before we can combine the variable terms, we need to get rid of that fraction.
First let’s put all the terms on the left over a common denominator of three:
Combining the fractions then gives us
3x+4−15x
3
Combining like terms in the numerator gives us
3x+4
3
− 15x
3 = 6.
= 6.
4−12x
3
= 6.
Multiplying both sides by 3 gives us 4 − 12x = 18.
Subtracting 4 from both sides gives us −12x = 14.
7
And finally, dividing both sides by -12 gives us x = − 14
12 , which reduces to x = − 6 .
Solving Multi-Step Equations Using the Distributive Property
You may have noticed that when one side of the equation is multiplied by a constant term, we can either distribute
it or just divide it out. If we can divide it out without getting awkward fractions as a result, then that’s usually the
better choice, because it gives us smaller numbers to work with. But if dividing would result in messy fractions, then
it’s usually better to distribute the constant and go from there.
1
www.ck12.org
Example B
Solve 7(2x − 5) = 21.
The first thing we want to do here is get rid of the parentheses. We could use the Distributive Property, but it just
so happens that 7 divides evenly into 21. That suggests that dividing both sides by 7 is the easiest way to solve this
problem.
If we do that, we get 2x − 5 =
then divide by 2 to get x = 4.
21
7
or just 2x − 5 = 3. Then all we need to do is add 5 to both sides to get 2x = 8, and
Example C
Solve 17(3x + 4) = 7.
Once again, we want to get rid of those parentheses. We could divide both sides by 17, but that would give us an
inconvenient fraction on the right-hand side. In this case, distributing is the easier way to go.
Distributing the 17 gives us 51x + 68 = 7. Then we subtract 68 from both sides to get 51x = −61, and then we
divide by 51 to get x = −61
51 . (Yes, that’s a messy fraction too, but since it’s our final answer and we don’t have to do
anything else with it, we don’t really care how messy it is.)
Example D
Solve 4(3x − 4) − 7(2x + 3) = 3.
Before we can collect like terms, we need to get rid of the parentheses using the Distributive Property. That gives
us 12x − 16 − 14x − 21 = 3, which we can rewrite as (12x − 14x) + (−16 − 21) = 3. This in turn simplifies to
−2x − 37 = 3.
Next we add 37 to both sides to get −2x = 40.
And finally, we divide both sides by -2 to get x = −20.
Watch this video for help with the Examples above.
MEDIA
Click image to the left for more content.
CK-12 Foundation: Multi-Step Equations
Vocabulary
• If dividing a number outside of parentheses will produce fractions, it is often better to use the Distributive
Property to expand the terms and then combine like terms to solve the equation.
Guided Practice
Solve the following equation for x: 0.1(3.2 + 2x) + 21 3 − 5x = 0
Solution:
2
www.ck12.org
Concept 1. Multi-Step Equations
This function contains both fractions and decimals. We should convert all terms to one or the other. It’s often easier
to convert decimals to fractions, but in this equation the fractions are easy to convert to decimals—and with decimals
we don’t need to find a common denominator!
In decimal form, our equation becomes 0.1(3.2 + 2x) + 0.5(3 − 0.2x) = 0.
Distributing to get rid of the parentheses, we get 0.32 + 0.2x + 1.5 − 0.1x = 0.
Collecting and combining like terms gives us 0.1x + 1.82 = 0.
Then we can subtract 1.82 from both sides to get 0.1x = −1.82, and finally divide by 0.1 (or multiply by 10) to get
x = −18.2.
Practice
Solve the following equations for the unknown variable.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
3(x − 1) − 2(x + 3) = 0
3(x + 3) − 2(x − 1) = 0
7(w + 20) − w = 5
5(w + 20) − 10w = 5
9(x − 2) − 3x = 3
12(t − 5) + 5 = 0
2(2d + 1)= 23
2 5a − 13 = 72
2
2
2
9 i+ 3 = 5
1
4 v + 4 = 35
2
g
6
=
10
3
m+3
m
1
2 − 4 = 3
5 3k + 2 = 32
3
3
2
z = 5
2
10
r +2 = 3
12
3+z
5 = z
3