Download 1 Multi-Step Equations with Like Terms

Multi-Step Equations with
Like Terms
Andrew Gloag
Eve Rawley
Anne Gloag
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Printed: October 25, 2013
AUTHORS
Andrew Gloag
Eve Rawley
Anne Gloag
www.ck12.org
C ONCEPT
Concept 1. Multi-Step Equations with Like Terms
1
Multi-Step Equations with
Like Terms
Here you’ll learn how to add and subtract like terms to simplify two-step equations and solve for their unknown.
What if you had an equation in which the same variable appeared twice, like 2(x − 4) + 4x = −23? How could you
simplify the equation so that the variable appears only once in order to solve for it? After completing this Concept,
you’ll be able to combine like terms to solve two-step equations like this one.
Watch This
MEDIA
Click image to the left for more content.
CK-12 Foundation: 0305S Combining Like Terms in Two-Step Equations
Guidance
When we look at a linear equation we see two kinds of terms: those that contain the unknown variable, and those
that don’t. When we look at an equation that has an x on both sides, we know that in order to solve it, we need to
get all the x−terms on one side of the equation. This is called combining like terms. The terms with an x in them
are like terms because they contain the same variable (or, as you will see in later chapters, the same combination of
variables).
TABLE 1.1:
Like Terms
x
4x, 10x, −3.5x, and 12
3y, 0.000001y, and y
xy, 6xy, and 2.39xy
Unlike Terms
3x and 3y
4xy and 4x
0.5x and 0.5
Example A
To add or subtract like terms, we can use the Distributive Property of Multiplication.
3x + 4x = (3 + 4)x = 7x
0.03xy − 0.01xy = (0.03 − 0.01)xy = 0.02xy
−y + 16y + 5y = (−1 + 16 + 5)y = 10y
5z + 2z − 7z = (5 + 2 − 7)z = 0z = 0
To solve an equation with two or more like terms, we need to combine the terms first.
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Example B
Solve (x + 5) − (2x − 3) = 6.
There are two like terms: the x and the −2x (don’t forget that the negative sign applies to everything in the
parentheses). So we need to get those terms together. The associative and distributive properties let us rewrite
the equation as x + 5 − 2x + 3 = 6, and then the commutative property lets us switch around the terms to get
x − 2x + 5 + 3 = 6, or (x − 2x) + (5 + 3) = 6.
(x − 2x) is the same as (1 − 2)x, or −x, so our equation becomes −x + 8 = 6
Subtracting 8 from both sides gives us −x = −2.
And finally, multiplying both sides by -1 gives us x = 2.
Example C
Solve 2x − 3x = 6.
This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator
of six.
3x 2x
−
=6
6
6
Then we subtract the fractions to get
x
6
= 6.
Finally we multiply both sides by 6 to get x = 36.
Watch this video for help with the Examples above.
MEDIA
Click image to the left for more content.
CK-12 Foundation: Combing Like Terms in 2 Step Equations
Vocabulary
• Terms with the same variable in them (or no variable in them) are like terms. Combine like terms (adding
or subtracting them from each other) to simplify the expression and solve for the unknown.
Guided Practice
Solve
2x
5
− 3x
2 = 11.
This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator
of ten.
4x 15x
−
= 11
10 10
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Concept 1. Multi-Step Equations with Like Terms
Then we subtract the fractions to get − 11x
10 = 11.
10
:
Finally we multiply both sides by − 11
10
10
− 11x
10 · − 11 = 11 · − 11
to get x = −10.
Practice
Solve the following equations for the unknown variable.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1.3x − 0.7x = 12
−10a − 2(a + 5) = 14
5(2y − 3y) = −20
2
1
14
3 x − 5 x = 15
5x − (3x + 2) = 1
5
s − 3s
8 = 6
10(y + 5y) = 10
2.3x + 2(0.75x − 3.5) = 7.5
3(x + 2) + 5(2 − x) = −32
6x + 2(5x − 2) = 12
3