Simplifying Algebraic Expressions

Learning Outcomes

  • Identify the variables and constants in a term
  • Identify the coefficient of a variable term
  • Identify and combine like terms in an expression


Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are
7,y,5x2,9a,and 13xy7,y,5{x}^{2},9a,\text{and }13xy
.

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term
3x3x
is
33
. When we write
xx
, the coefficient is
11
, since
x=1xx=1\cdot x
. The table below gives the coefficients for each of the terms in the left column.

Term Coefficient
77
77
9a9a
99
yy
11
5x25{x}^{2}
55
An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Expression Terms
77
77
yy
yy
x+7x+7
x,7x,7
2x+7y+42x+7y+4
2x,7y,42x,7y,4
3x2+4x2+5y+33{x}^{2}+4{x}^{2}+5y+3
3x2,4x2,5y,33{x}^{2},4{x}^{2},5y,3

example

Identify each term in the expression
9b+15x2+a+69b+15{x}^{2}+a+6
. Then identify the coefficient of each term.

Solution:

The expression has four terms. They are
9b,15x2,a9b,15{x}^{2},a
, and
66
.

  • The coefficient of
    9b9b
    is
    99
    .
  • The coefficient of
    15x215{x}^{2}
    is
    1515
    .
  • Remember that if no number is written before a variable, the coefficient is
    11
    . So the coefficient of
    aa
    is
    11
    .
  • The coefficient of a constant is the constant, so the coefficient of
    66
    is
    66
    .


 

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Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

5x,7,n2,4,3x,9n25x,7,{n}^{2},4,3x,9{n}^{2}


Which of these terms are like terms?

  • The terms
    77
    and
    44
    are both constant terms.
  • The terms
    5x5x
    and
    3x3x
    are both terms with
    xx
    .
  • The terms
    n2{n}^{2}
    and
    9n29{n}^{2}
    both have
    n2{n}^{2}
    .


Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms
5x,7,n2,4,3x,9n25x,7,{n}^{2},4,3x,9{n}^{2}
,

  • 77
    and
    44
    are like terms.
  • 5x5x
    and
    3x3x
    are like terms.
  • n2{n}^{2}
    and
    9n29{n}^{2}
    are like terms.


Like Terms

Terms that are either constants or have the same variables with the same exponents are like terms.

example

Identify the like terms:

  1. y3,7x2,14,23,4y3,9x,5x2{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}
  2. 4x2+2x+5x2+6x+40x+8xy4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy






 

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Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think
3x+6x3x+6x
would simplify to? If you thought
9x9x
, you would be right!

We can see why this works by writing both terms as addition problems.

The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x. Add the coefficients and keep the same variable. It doesn’t matter what
xx
is. If you have
33
of something and add
66
more of the same thing, the result is
99
of them. For example,
33
oranges plus
66
oranges is
99
oranges. We will discuss the mathematical properties behind this later.

The expression
3x+6x3x+6x
has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.

The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together. Now it is easier to see the like terms to be combined.

Combine like terms

  1. Identify like terms.
  2. Rearrange the expression so like terms are together.
  3. Add the coefficients of the like terms.


 

example

Simplify the expression:
3x+7+4x+53x+7+4x+5
.





 

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example

Simplify the expression:
8x+7x2x2+4x8x+7{x}^{2}-{x}^{2}-+4x
.





 

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In the following video, we present more examples of how to combine like terms given an algebraic expression.



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