4.4: Combining Like Terms

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May 28, 2024
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- 4.3: Simplifying Algebraic Expressions
- 5: Solving Equations

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We begin our discussion with the definition of a term.

Definition: Term

A term is a single number or variable, or it can be the product of a number (called its coefficient) and one or more variables (called its variable part). The terms in an algebraic expression are separated by addition symbols.

Example 1

Identify the terms in the algebraic expression

$3x^2 + 5xy + 9y^2 + 12\nonumber$

For each term, identify its coefficient and variable part.

Solution

In tabular form, we list each term of the expression 3x^2 + 5xy + 9y^2 + 12, its coefficient, and its variable part.

Term	Coefficient	Variable Part
3x²	3	x²
5xy	5	xy
9y²	9	y²
12	12	None

Exercise

How many terms are in the algebraic expression 3x² + 2xy − 3y²?

Answer: 3

Example 2

Identify the terms in the algebraic expression

$a^3 − 3a^2b + 3ab^2 − b^3\nonumber$

For each term, identify its coefficient and variable part.

Solution

The first step is to write each difference as a sum, because the terms of an expression are defined above to be those items separated by addition symbols.

$a^3 + (−3a^2b)+3ab^2 + (−b^3)\nonumber$

In tabular form, we list each term of the expression $a^3+(−3a^2b)+3ab^2+(−b^3)$ , its coefficient, and its variable part.

Term	Coefficient	Variable Part
a³	1	a³
-3a²b	−3	a²b
3ab²	3	ab²
−b³	−1	b³

Exercise

How many terms are in the algebraic expression $11 − a^2 − 2ab + 3b^2a$ ?

Answer: 4

Like Terms

We define what is meant by “like terms” and “unlike terms.”

Definition

Like and Unlike Terms. The variable parts of two terms determine whether the terms are like terms or unlike terms.

Like Terms. Two terms are called like terms if they have identical variable parts, which means that the terms must contain the same variables raised to the same exponential powers.

Unlike Terms. Two terms are called unlike terms if their variable parts are different.

Example 3

Classify each of the following pairs as either like terms or unlike terms: (a) 3x and −7x, (b) 2y and 3y², (c) −3t and 5u, and (d) −4a³ and 3a³.

Solution

Like terms must have identical variable parts.

3x and −7x have identical variable parts. They are “like terms.”
2y and 3y² do not have identical variable parts (the exponents differ). They are “unlike terms.”
−3t and 5u do not have identical variable parts (different variables). They are “unlike terms.”
−4a³ and 3a³ have identical variable parts. They are “like terms.”

Exercise

Are −3xy and 11xy like or unlike terms?

Answer: Like terms

Combining Like Terms

When using the distributive property, it makes no difference whether the multiplication is on the left or the right, one still distributes the multiplication times each term in the parentheses.

Distributive Property

If a, b, and c are integers, then

a(b + c) = ab + ac and (b + c)a = ba + ca.

In either case, you distribute a times each term of the sum.

“Like terms” can be combined and simplified. The tool used for combining like terms is the distributive property. For example, consider the expression 3y + 7y, composed of two “like terms” with a common variable part. We can use the distributive property and write

$3y+7y=(3+7)y\nonumber$

Note that we are using the distributive property in reverse, “factoring out” the common variable part of each term. Checking our work, note that if we redistribute the variable part y times each term in the parentheses, we are returned to the original expression 3y + 7y.

Example 4

Use the distributive property to combine like terms (if possible) in each of the following expressions: (a) −5x² − 9x², (b) −5ab + 7ab, (c) 4y³ − 7y², and (d) 3xy² − 7xy².

Solution

If the terms are “like terms,” you can use the distributive property to “factor out” the common variable part.

a) Factor out the common variable part x².

$\begin{aligned} -5x^2 -9x^2 =(-5-9)x^2 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ = -14x^2 ~ & \textcolor{red}{ \text{ Simplify: } -5-9=-5+(-9) = -14.} \end{aligned}\nonumber$

b) Factor out the common variable part ab.

$\begin{aligned} -5ab +7ab = (-5+7)ab ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =2ab ~ & \textcolor{red}{ \text{ Simplify: } -5+7 = 2.} \end{aligned}\nonumber$

c) The terms in the expression 4y³ − 7y² have different variable parts (the exponents are different). These are “unlike terms” and cannot be combined.

d) Factor out the common variable part xy².

$\begin{aligned} 3xy^2 - 7xy^2 =(3-7)xy^2 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =-4xy^2 ~ & \textcolor{red}{ \text{ Simplify: } 3-7=3+(-7)=-4.} \end{aligned}\nonumber$

Exercise

Simplify: −8z − 11z

Answer: −19z

Speeding Things Up a Bit

Once you’ve written out all the steps for combining like terms, like those shown in Example 4, you can speed things up a bit by following this rule:

Combining Like Terms

To combine like terms, simply add their coefficients and keep the common variable part.

Thus for example, when presented with the sum of two like terms, such as in 5x+ 8x, simply add the coefficients and repeat the common variable part; that is, 5x + 8x = 13x.

Example 5

Combine like terms:

−9y − 8y,
−3y⁵ + 4y⁵ and
−3u² + 2u².

Solution

a) Add the coefficients and repeat the common variable part. Therefore,

$−9y − 8y = −17y.\nonumber$

b) Add the coefficients and repeat the common variable part. Therefore,

$−3y^5 + 4y^5 = 1y^5.\nonumber$

However, note that 1y⁵ = y⁵. Following the rule that the final answer should use as few symbols as possible, a better answer is −3y⁵ + 4y⁵ = y⁵.

c) Add the coefficients and repeat the common variable part. Therefore,

$−3u^2 + 2u^2 = (−1)u^2.\nonumber$

However, note that (−1)u² = −u². Following the rule that the final answer should use as few symbols as possible, a better answer is −3u²+ 2u² = −u².

Simplify

A frequently occurring instruction asks the reader to simplify an expression.

Simplify

The instruction simplify is a generic term that means “try to write the expression in its most compact form, using the fewest symbols possible.”

One way you can accomplish this goal is by combining like terms when they are present.

Example 6

Simplify: 2x + 3y − 5x + 8y.

Solution

Use the commutative property to reorder terms and the associative and distributive properties to regroup and combine like terms.

$\begin{aligned} 2x + 3y - 5x + 8y = (2x - 5x) + (3y + 8y) ~ & \textcolor{red}{ \text{ Reorder and regroup.}} \\ = -3x + 11y ~ & \textcolor{red}{ \text{ Combine like terms:}} \\ ~ & \textcolor{red}{ 2x - 5x = -3x \text{ and } 3y + 8y = 11y.} \end{aligned}\nonumber$

Alternate solution

Of course, you do not need to show the regrouping step. If you are more comfortable combining like terms in your head, you are free to present your work as follows:

$2x + 3y − 5x + 8y = −3x + 11y.\nonumber$

Exercise

Simplify: −3a + 4b − 7a − 9b

Answer: −10a − 5b

Example 7

Simplify: −2x − 3 − (3x + 4).

Solution

First, distribute the negative sign.

$\begin{aligned} -2x-3-(3x+4)= -2x-3-3x-4 ~ & \textcolor{red}{-(3x+4)=-3x-4.} \end{aligned}\nonumber$

Next, use the commutative property to reorder, then the associative property to regroup. Then combine like terms.

$\begin{aligned} =(-2x-3x)+(-3-4) ~ & \textcolor{red}{ \text{ Reorder and regroup.}} \\ =-5x+(-7) ~ & \textcolor{red}{ \text{ Combine like terms:}} \\ ~ & \textcolor{red}{ -2x-3x=-5x.} \\ =-5x-7 ~ & \textcolor{red}{ \text{ Simplify:}} \\ ~ & \textcolor{red}{-5x+(-7)=-5x-7.} \end{aligned}\nonumber$

Alternate solution

You may skip the second step if you wish, simply combining like terms mentally. That is, it is entirely possible to order your work as follows:

$\begin{aligned} -2x-3-(3x+4) = -2x-3-3x-4 ~ & \textcolor{red}{ \text{ Distribute negative sign.}} \\ =-5x-7 ~ & \textcolor{red}{ \text{ Combine like terms.}} \end{aligned}\nonumber$

Exercise

Simplify: −9a − 4 − (4a − 8)

Answer: −13a + 4

Example 8

Simplify: 2(5 − 3x) − 4(x + 3).

Solution

Use the distributive property to expand, then use the commutative and associative properties to group the like terms and combine them.

\[ \begin{aligned} 2(5-3x)-4(x+3) = 10-6x-4x-12 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =(-6x-4x)+(10-12) ~ & \textcolor{red}{ \text{ Group like terms.}} \\ =-10x-2 ~ & \textcolor{red}{ \text{ Combine like terms: }} \\ ~ & \textcolor{red}{-6x-4x=-10x \text{ and} \\ ~ & \textcolor{red}{10-12=-2.} \end{aligned}\nonumber \]

Alternate solution

You may skip the second step if you wish, simply combining like terms mentally. That is, it is entirely possible to order your work as follows:

$\begin{aligned} 2(5-3x)-4(x+3) = 10-6x-4x-12 ~ & \textcolor{red}{ \text{ Distribute.}} \\ =-10x-2 ~& \textcolor{red}{ \text{ Combine like terms.}} \end{aligned}\nonumber$

Exercise

Simplify: −2(3a − 4) − 2(5 − a)

Answer: −4a − 2

Example 9

Simplify: −8(3x²y − 9xy) − 8(−7x² − 8xy)

Solution

We will proceed a bit quicker with this solution, using the distributive property to expand, then combining like terms mentally.

$\begin{aligned} -8(3x^2y-9xy) -8(-7x^2y-8xy)=-24x^2y+72xy+56x^2y+64xy \\ = 32x^2y+136xy \end{aligned}\nonumber$

Exercise

Simplify: (a² − 2ab) − 2(3ab + a²)

Answer: −a² − 8ab

Exercises

Combine like terms and simplify.

1. 17xy² + 18xy² + 20xy²

7. 12r − 12r

8. 16s − 5s

11. −5q + 7q

13. r − 13r − 7r

14. 19m + m + 15m

16. 13x²y + 2x²y

17. −8 + 17n + 10 + 8n

19. −2x³ − 19x²y − 15x²y + 11x³

23. −13 + 16m + m + 16

29. 9n + 10 + 7 + 15n

31. 3y +1+6y + 3

35. 3(−4x² + 10y²) + 10(4y² − x²)

39. −10q − 10 − (−3q + 5)

41. 7(8y + 7) − 6(8 − 7y)

45. −2(6 + 4n) + 4(−n − 7)

47. 8 − (4 + 8y)

53. 7(1 + 7r) + 2(4 − 5r)

55. −2(−5 − 8x²) − 6(6)

Answers

1. 55xy²

7. 0

8. 11s

11. 2q

13. −19r

14. 35m

16. 15x²y

17. 2 + 25n

19. 9x³ − 34x²y

23. 3 + 17m

29. 24n + 17

31. 9y + 4

35. −22x² + 70y²

39. −7q − 15

41. 98y + 1

45. −40 − 12n

47. 4 − 8y

53. 15 + 39r

55. −26 + 16x²

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