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Section 1.2 Combining Like Terms

Algebraic expressions can be large and complicated, so anything we can do to rewrite an expression in a simpler way is helpful. One of the fundamental skills we have for simplifying expressions is combining like terms.
Figure 1.2.1. Alternative Video Lesson

Subsection 1.2.1 Identifying Terms

Definition 1.2.2.

In an algebraic expression, the terms are quantities being added together.
Note that terms are not the same thing as factors, which are quantities being multiplied together. Factors are studied more later in this book.

Example 1.2.3.

List the terms in the expression 2ℓ+2w.
Explanation.
The expression has two terms that are being added, 2ℓ and 2w.
When an expression has subtraction, we can rewrite the it using addition of a negative to make it easier to see exactly what the terms are.

Example 1.2.4.

List the terms in the expression −3x2+5x−4.
Explanation.
We can rewrite this expression as −3x2+5x+(−4) to see that the terms are −3x2, 5x, and −4. Note that the third term is −4, not just 4.

Example 1.2.5.

List the terms in the expression 3cm+2cm−3cm+2cm.
Explanation.
This expression has four terms: 3 cm, 2 cm, −3 cm, and 2 cm.

Checkpoint 1.2.6.

List the terms in the expression 5x−4x+10z.
Explanation.
The terms are 5x, −4x, and 10z.

Subsection 1.2.2 Combining Like Terms

If you have 3cm+2cm, it is natural to add those together to get 5 cm. That works because their units (cm) are the same. The same idea applies to other terms, even ones that don’t have units. For example, with 2x+3x, we have 2 things and then 3 more of those things. All together, we have 5 of those things. So 2x+3x is the same as 5x.
Terms in an algebraic expression that can be combined by adding them together into one new term are called like terms.
  • Sometimes like terms use a common variable, like how 2x+3x has terms that each use x. This simplifies to 5x.
  • Sometimes like terms use the same units, like how 3cm+2cm has terms that each use cm. This simplifies to 5 cm.
  • Sometimes like terms have something else in common, like how 37+27 has terms that each use 7. This simplifies to 57.

Example 1.2.7.

In the expressions below, look for like terms and then simplify where possible by adding or subtracting.
  1. 5in+20in
  2. 16ft2+4ft
  3. 2🍎+5🍎
  4. 5min+12ft
  5. 5🐶−2🐱
  6. 20m−6m
Explanation.
We can combine terms with the same units, but we cannot combine terms with distinct units such as minutes and feet.
  1. 5in+20in=25in
  2. 16ft2+4ft cannot be combined
  3. 2🍎+5🍎=7🍎
  4. 5min+12ft cannot be combined
  5. 5🐶−2🐱 cannot be combined
  6. 20m−6m=14m
One of the examples from Example 7 was 16ft2+4ft. The units on these two terms look similar, but they are different. 16 ft2 is a measurement of how much area something has. 4 ft is a measurement of how long something is. Figure 8 illustrates this.
a four-by-four square, illustrating 16 square feet, alongside a line that is four feet long
Figure 1.2.8. There is no way to add 16 ft2 to 4 ft.

Example 1.2.9.

Simplify each expression by combining like terms (if possible).
  1. 57x−27x+37y
  2. 12x+12x2
  3. −3.01r+1.2s−5t
  4. 23y+56y
  5. 103t−12x−t
  6. x−0.15x
Explanation.
  1. This expression has two like terms, 57x and −27x, which we can combine. We have to subtract 57−27, which is straightforward since they have the same denominator.
    57x−27x+37y=37x+37y
    We don’t combine 37x and 37y because x and y are different variables.
  2. This expression cannot be simplified because the variable parts are not the same. We cannot add x-terms with x2-terms just like we cannot add feet (a measure of length) with square feet (a measure of area).
  3. There aren’t any like terms.
  4. The two terms are like terms. To combine them, we need to add 23+56. For a review of adding fractions with different denominators, see Section A.2. In this case,
    23+56=23⋅22+56=46+56=96=32
    So 23y+56y=32y.
  5. There are two like terms: 103t and −t. A lonely t is usually just written as t, but we can think of it as 1t. So we are combining 103t and −t and we need to subtract 103−1.
    103−1=103−33=73
    Our two terms combine to make 73t. There was another term when this started and the final simplified expression is 73t−12x.
  6. This expression can be thought of as 1.00x−0.15x. Subtracting decimals 1.00−0.15, the result is 0.85. So we have 0.85x.

Checkpoint 1.2.10.

Simplify each expression by combining like terms (if possible).
(a)
x+0.25x
Explanation.
Think of this expression as 1.00x+0.25x and simplify to get 1.25x.
(b)
49x−710y+23x
Explanation.
This expression has two like terms that can be combined: 49x and 23x. To combine them, we need to add the fractions 49+23.
49+23=49+69=109
So 49x+23x=109x. And together with the third term, the answer is 109x−710y.
(c)
56y−815y+23y2
Explanation.
In this expression we can combine the y-terms. We need to subtract the fractions 56−815.
56−815=2530−1630=930=310
So 56y−815y=310y. And together with the third term, the answer is 310y+23y2.
(d)
4x+1.5y−9z
Explanation.
This expression cannot be simplified further because there are not any like terms.

Subsection 1.2.3 Applications

The perimeter of a shape is the length of a strip of tape that could be taped tightly around the shape. When a shape has straight side edges, the perimeter comes from adding all of the side lengths together. This can lead to like terms that can be combined.

Example 1.2.11.

Find the perimeter of this shape, which is not drawn to scale. Simplify the perimeter expression as much as possible.
a five-sided shape whose sides are labeled 2x, 3y, 1.5x, 5y, and x
Explanation.
The perimeter is the result from adding the five sides together: 2x+3y+1.5x+5y+x. There are three x-terms that sum to 4.5x, and two y-terms that sum to 8y. So the perimeter is 4.5x+8y.

Checkpoint 1.2.12.

Find the perimeter of this shape, which is not drawn to scale. Simplify the perimeter expression as much as possible.
a six-sided shape whose sides are labeled 3/2 A, 2/3 B, 3/4 C, 2A, 2/3 B, and 8/17 C
Explanation.
The perimeter is the result from adding the six sides together: 32A+23B+34C+2A+23B+817C. There are two A-terms, two B-terms, and two C-terms. We need to find:
32+2=32+21⋅2223+23=4334+817=34⋅1717+81744=32+42=5168+3268=72=8368
So the perimeter is 72A+43B+8368C.
Sometimes it makes sense to add two algebra expressions together, and that may give an opportunity to combine like terms.

Example 1.2.13.

A chemist has a bottle with 1.2 L of water mixed with 0.3 L of gasoline. At this time, they have forgotten the density of water (in g⁄L) and the density of gasoline (also in g⁄L) so they use w and g as variables for these densities. This means 1.2w+0.3g is the total mass of this mixture, in grams.
There is a second bottle, with 0.9 L of water mixed with 0.5 L of gasoline. The chemist pours it all together. What is the mass of the combined mixture?
Explanation.
The first mass is 1.2w+0.3g grams, and the second mass is 0.9w+0.5g grams. So together, the mass in grams is:
(1.2w+0.3g)+(0.9w+0.5g)=1.2w+0.3g+0.9w+0.5g=2.1w+0.8g

Reading Questions 1.2.4 Reading Questions

1.

What should you be aware of when there is subtraction in an algebraic expression and you are identifying its terms?

2.

Describe at least two different ways in which a pair of terms would be considered to be “like terms”.

3.

What might help when you are combining like terms, and one of the terms is a lonely variable like x or y?

Exercises 1.2.5 Exercises

Prerequisite/Review Skills

These exercises are only intended for students who are rusty with adding/subtracting decimals and fractions. If you feel comfortable, proceed to Skills Practice.
Fraction Arithmetic.
Add or subtract the fractions.
1.
717+1517
2.
417+917
3.
1317−917
4.
1419−1319
5.
716+1516
6.
518+1718
7.
712−512
8.
1116−316
9.
411+7
10.
513+3
11.
1−710
12.
35−92
13.
513+114
14.
115+919
15.
34−23
16.
13−310
17.
89+563
18.
23+1121
19.
25−340
20.
45−1135
21.
14+512
22.
13+715
23.
712−14
24.
18−588
25.
320+76
26.
710+435
27.
109−421
28.
914−58
29.
736+528
30.
56+914
31.
514−310
32.
56−215

Skills Practice

Identifying Terms.
List the terms in each expression.
33.
−12u+t+2.7D+18b
34.
−85c−Q−15p+3.9b
35.
310J−13k+49J
36.
−83s−45G+53G
37.
110+77x5−x9
38.
81z4−2.1z+4.2z9
39.
5.5y8−y9+y9
40.
−60t2+63t6+t6+t2
Combining Like Terms.
Simplify the expression by combining like terms if possible.
41.
8.8K+9.4K
42.
36R−9.2R
43.
X+15X
44.
110c+34c
45.
8.1i−l
46.
73n+5T
47.
−10u+4u3+83u4
48.
−6z+107z3+3.9z4
49.
12R−53F−72F
50.
16K−18y+12K
51.
0.4g+6.4g+34R
52.
1.8W−4W+6.2N
53.
109c2+14c2−2c5
54.
−92h6−45h+h6
55.
0.6n4+5.2n6+9.7n6
56.
1.5u3+9.2u3−2.8u2
57.
−7z−52z+12z
58.
52F+8F+34F
59.
−K−4.2K+1.3K
60.
5.1R+R−5.4R
61.
43W+2H+73W+910H
62.
83r+87c−92r−53c
63.
7.2Y−7.1h+0.8h+3.3Y
64.
8.6G+48G−80n+n
65.
0.3t6+t6−81t2+9.8t2
66.
−9.4z2+69z5+z2−9.1z5
67.
67E3−35E5+103E3+E5
68.
310K3−49K4+23K3+K4

Applications

Perimeter.
Write a simplified expression for the perimeter of the given shape (which is not drawn to scale).
69.
a quadrilateral whose sides are labeled (1/4*U), (7/4*Q), (2*U), and (8/3*Q).
70.
a quadrilateral whose sides are labeled (5/7*B), (7/8*W), (6/5*W), and (4/3*B).
71.
a quadrilateral whose sides are labeled (k), (c), (c), (2.3*k), (4.9*c), and (78*k).
72.
a quadrilateral whose sides are labeled (0.1*R), (h), (h), (R), (13*R), and (83*h).
73.
a quadrilateral whose sides are labeled (1/2*c), (5/8*n), (3/2*c), and (3/2*z).
74.
a quadrilateral whose sides are labeled (3/10*g), (1/3*t), (7/9*z), and (9/8*z).
75. Office Lunch.
Every Friday, an office supervisor provides catered lunches for everyone working in the office. People can order a sandwich (costs S dollars) or a burrito (costs B dollars).
One week, the office ordered 13 sandwiches and 10 burritos. So the bill came to 13S+10B dollars.
(a)
The next week, the office ordered 14 sandwiches and 8 burritos. How much was the bill that week, in dollars?
(b)
What was the combined bill for these two weeks?
76. Pies and Cakes.
Kaylin and Humberto are co-owners of a pastry shop. Kaylin bakes pies and Humberto bakes cakes. Kaylin is able to bake p pies each day she works and Humberto is able to bake c cakes each day he works.
One month, Kaylin worked 18 days and Humberto worked 20 days. That month they produced 18p+20c baked goods in total.
(a)
The next month, Kaylin worked 22 days and Humberto worked 19 days. How many baked goods did they produce that month?
(b)
How many baked goods was that in total for those two months?
77. Sportsball.
Sportsball is a game similar to basketball. There are three ways for a team to score points: a “long field goal” earns L points; a “short field goal” earns S points; and a “penalty shot” earns P points.
In the first half of one game, a team scored 6 penalty shots, 24 long field goals, and 37 short field goals. So they earned 6P+24L+37S points.
(a)
In the second half of the game, that same team scored 19 penalty shots, 36 long field goals, and 44 short field goals. How many points did they earn during the second half?
(b)
How many points did the team score in total?
78. Metal Recycling.
A metals recycling plant makes its revenue from selling aluminum at A dollars per ton, steel at S dollars per ton, and tin at T dollars per ton. One week, they processed 3.64 tons of aluminum, 2.28 tons of steel, and 2.91 tons of tin. So they generated 3.64A+2.28S+2.91T dollars in revenue.
(a)
The next week, they processed 3.5 tons of aluminum, 2.17 tons of steel, and 2.72 tons of tin. How much revenue (in dollars) did this generate that week?
(b)
How much revenue (in dollars) was that in total those two weeks?
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