Chapter 2: Linear Equations and Inequalities
2.3: Solving Equations with Like Terms
When isolating the variable in the equation, it may be necessary to first combine like
terms.
Definitions
A term is a number {1,2,3,…}, a variable {x,y,z,a,b,c…}, or the product of a number and
a variable {2x, 3y, ½ a, etc.). Terms are separated by + or – signs in an expression, and
the + or – signs are part of each term. (Everything inside parenthesis is treated as one
term until the parentheses are removed.)
A variable is a letter that represents an unknown value(s). When we are asked to solve
an equation, it usually means that we must isolate the variable and find its value.
A coefficient is a number that comes in front of a variable. A coefficient can be an
integer, a decimal, or a fraction. A coefficient multiplies the variable. Every variable has
a coefficient. If a variable appears to have no coefficient, it’s coefficient is an “invisible
1”
Like terms are terms with exactly the same variable. Like terms must have identical
exponents. Like terms can be combined by adding their coefficients.
Examples of Like Terms:
3x and 4x are like terms since they have exactly the same variable. They can be
combined into one variable by adding their coefficients.

There are two methods for combining like terms in the same expression, as
follows:
Method #1
Method #2
Line the like terms up in columns like an
Use factoring (the distributive property in
addition problem and add the coefficients:
reverse) to create a parenthetical grouping
3x
of the coefficients:
3x  4 x 
+4x
3  4 x 
7x
7x




3x and 4x 2 are not like terms since the have different exponents.
x and y are not like terms since they have different variables.
3 and 4 are like terms and may be combined since they are ordinary numbers.
3x and 4 are not like terms since one term is a variable term and the other term is
an ordinary number.
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Chapter 2: Linear Equations and Inequalities
2.3: Solving Equations with Like Terms
Ordering Terms. When combining like terms, it is often desirable to put terms in
alphabetical order, with decreasing exponents and with ordinary numbers last.
Example.
7-5a-c+8a+2+3c-7d+a=
-5a+8a+a-c+3c-7d+7+2=
(note that there are 8 terms)
(put in alpha order and
check to make sure there are still 8 terms)
 -5+8+1 a+  -1+3 c+  -7d  +  7+2  =
(identify the coefficients for adding)
4a+2c-7d+9
(simplify)
When an equation has like terms in different expressions, you can use the inverse
operations of addition or subtraction to move the terms into the same expression. It is
usually better to move the smaller like term rather than the larger like term because
moving the smaller like term typically results in a positive rather than a negative variable
term.
Example
Given:
2x+4 = x+7
Subtract  x  -x
-x
Subtract  4
Answer
Check
x+4 =
-4
7
-4
x
= 3
2(3)+4 = 3+7
6+4 = 10
10 = 10
Work on the following problems in your groups:
What is the solution of the equation 3y – 5y + 10 = 36?
(1) –13
(3) 4.5
(2) 2
(4) 13
Given:
CLT
S10
3y - 5y + 10 = 36
 3-5  y+10=36
-2y+10=36
-2y=36-10
-2y=26
D2
y= -13
Check:
3  -13 - 5  -13 + 10 = 36
-39+65+10=36
26+10=36
36=36
The answer is choice (4) 13.
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Chapter 2: Linear Equations and Inequalities
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2.3: Solving Equations with Like Terms
Many of the problems in this category require that you write and solve a simple equation
with a single variable. What follows is an example.
At a concert, $720 was collected for hot dogs, hamburgers, and soft drinks. All three
items sold for $1.00 each. Twice as many hot dogs were sold as hamburgers. Three times
as many soft drinks were sold as hamburgers. The number of soft drinks sold was
(1) 120
(3) 360
(2) 240
(4) 480
We can start by writing 720=hot dogs + hamburgers + soft drinks.
 Everything seems to be compared to the number of hamburgers, so let x = the
number of hamburgers.
 Twice as many hot dogs were sold as hamburgers, so let 2x = the number of hot
dogs.
 Three times as many soft drinks were sold as hamburgers, so let 3x = the number
of soft drinks.
We can now write 720=2x+x+3x. Now we have to solve our first degree equation.
Given
720=2x+x+3x
CLT
720=(2+1+3)x
D 6 
720=6x
120=x
We are not quite finished. The problem asks for the number of soft drinks that were sold.
We have solved for x, but x is the number of hamburgers that were sold. We said “let 3x
= the number of soft drinks,” so we have to multiply 120 times 3. The number of soft
drinks sold is 120  3  360 .
The answer is choice (3) 360.
We can check our work as follows:
Given
720=2x+x+3x
x=120
720=2 120  + 120  +3 120 
720=240+120+360
720=360+360
720=720
3
What is the solution for the equation
x  1  x  2?
(1) -1
(3) all real numbers
1
(2)
(4) There is no
2
solution.
(4)
x 1  x  2
1 2
Chapter 2: Linear Equations and Inequalities
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2.3: Solving Equations with Like Terms
4
What is the value of x in the
equation 2( x  3)  1  19 ?
(1) 6
(3) 10.5
(2) 9
(4) 12
5
What is the value of m in the
equation 2m  (m  1)  0 ?
1
(1) 1
(3)
3
(2) -1
(4) 0
6
Which value of p is the solution of
5 p  1  2 p  20 ?
19
(1)
(3) 3
7
19
(2)
(4) 7
3
7
If 3( x  2)  2( x  1)  8, the value
of x is
(1) 1
(3) 5
1
(2)
(4) 4
5
8
What is the value of x in the
equation 5  3x  7 ?
2
(1) 
(3) -4
3
2
(2)
(4) 4
3
9
What is the value of p in the
equation 8 p  2  4 p  10 ?
(1) 1
(3) 3
(2) -1
(4) -3
Solve for g: 3  2 g  5g  9
10
(4)
2( x  3)  1  19
2 x  6  18
2 x  24
x  12
(1)
2m  (m  1)  0
2m  m  1  0
m1
(4)
5 p  1  2 p  20
3 p  21
p7
(4)
3( x  2)  2( x  1)  8
3x  6  2 x  2  8
x4 8
x4
(4)
5  3 x  7
3x  12
x4
(4)
8 p  2  4 p  10
4 p  12
p  3
Chapter 2: Linear Equations and Inequalities
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2.3: Solving Equations with Like Terms
3  2 g  5g  9
12  3g
4g
11
Solve for x: 5( x  2)  2(10  x)
12
What is the value of x in the
equation 6( x  2)  36  10x ?
(1) -6
(3) 3
(2) 1.5
(4) 6
5( x  2)  2(10  x )
5x  10  20  2 x
3x  30
x  10
(3)
6( x  2)  36  10 x
6 x  12  36  10 x
16 x  48
x3
13
14
15
What is the value of n in the
equation 3n  8  32  n ?
(1) -10
(3) 6
(2) -6
(4) 10
What is the value of p in the
equation 2(3 p  4)  10 ?
(1) 1
(3) 3
1
1
(2) 2
(4)
3
3
What is the value of x in the
equation 13x  2( x  4)  8 x  1?
(1) 1
(3) 3
(2) 2
(4) 4
(4)
3n  8  32  n
4n  40
n  10
(3)
2(3 p  4)  10
6 p  8  10
6 p  18
p3
(3)
13x  2( x  4)  8 x  1
13x  2 x  8  8 x  1
11x  8 x  9
3x  9
x3
16
What is the value of x in the
equation 5(2 x  7)  15 x  10 ?
(1) 1
(3) -5
(2) 0.6
(4) -9
(3)
Chapter 2: Linear Equations and Inequalities
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2.3: Solving Equations with Like Terms
5(2 x  7)  15 x  10
10 x  35  15 x  10
25  5 x
x  5
17
18
19
If 2 x  3  7 and 3x  1  5  y, the
value of y is
(1) 1
(3) 10
(2) 0
(4) 10
If 3(x – 2) = 2x + 6, the value of x is
(1) 0
(3) 12
(2) 5
(4) 20
If 2(x + 3) = x + 10, then x equals
(1) 14
(3) 5
(2) 7
(4) 4
(3)
2 x  3  7
2 x  4
x  2
3( 2)  1  5  y
5  5  y
y  10
(3)
3( x  2)  2 x  6
3x  6  2 x  6
x  12
(4)
2( x  3)  x  10
2 x  6  x  10
x4
20
If 2x + 5 = –25 and –3m – 6 = 48,
what is the product of x and m?
(1) –270
(3) 3
(2) –33
(4) 270
2 x  5  25
2 x  30
x  15
(4)
3m  6  48
3m  54
m  18
The product of x and m is 270.
21
22
What is the solution of the equation
3 y  5 y  10  36 ?
(1) –13
(3) 4.5
(2) 2
(4) 13
Solve for x: 15x – 3(3x + 4) = 6
(1) 1
(3) 3
1
1
(2) 
(4)
2
3
(1)
3 y  5 y  10  36
2 y  26
y  13
(3)
15 x  3(3x  4)  6
15 x  9 x  12  6
6 x  18
x3
Chapter 2: Linear Equations and Inequalities
2.3: Solving Equations with Like Terms
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