Chapter 3: Variables, Expressions and Equations

3.2.1: Simplifying Expressions

Learning Outcomes

  • Identify the coefficient of a variable term
  • Recognize and combine like terms in an expression
  • Use the order of operations to simplify expressions containing like terms

Key words

  • Term: a constant or the product or quotient of constants and one or more variables
  • Constant: a number that has a fixed value
  • Variable: a letter that represents a value that can change
  • Algebraic expression: is a term, or the sum or difference of terms
  • Coefficient: the number in front of a variable or term
  • Like terms: terms that have the same variables and exponents
  • Commutative Property of Addition: we can change the order of addends without changing the sum
  • Associative Property of Addition: we can regroup the addends without changing the sum
  • Commutative Property of Multiplication: we can change the order of the factors without changing the product
  • Associative Property of Multiplication: we can regroup the factors without changing the product
  • Distributive Property of Multiplication over Addition: a constant is multiplied onto an expression by multiplying each term in the expression by the constant

Identify Terms, Coefficients, and Like Terms

In mathematics, we may see expressions such as [latex]x+5[/latex],  [latex]\dfrac{4}{3}{r}^{3}[/latex], or [latex]5m-2n+6mn[/latex]. Algebraic expressions are made up of terms.

A term is a constant or the product or quotient of constants and one or more variables. Some examples of terms are [latex]7,y,5{x}^{2},-\frac{9a}{2b},\text{ and }-13xy[/latex].

In the expression [latex]x+5[/latex], [latex]5[/latex] is called a constant because it does not vary and [latex]x[/latex] is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a term, or the sum of terms (remember that subtraction can always be written as a sum of the opposite).

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 [latex]3x[/latex] is [latex]3[/latex]. When we write [latex]x[/latex], the coefficient is [latex]1[/latex], since [latex]x=1\cdot x[/latex]. The table below gives the coefficients for each of the terms in the left column.

Term Coefficient
[latex]7[/latex] [latex]7[/latex]
[latex]-9a[/latex] [latex]-9[/latex]
[latex]y[/latex] [latex]1[/latex]
[latex]-\frac{2}{5}{x}^{2}[/latex] [latex]-\frac{2}{5}[/latex]

The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation with the term when we list it. Think of the operation as belonging to the term it precedes.

Expression Terms
[latex]-7[/latex] [latex]-7[/latex]
[latex]y[/latex] [latex]y[/latex]
[latex]x-\frac{7}{5}[/latex] [latex]x,-\frac{7}{5}[/latex]
[latex]2x+7y-4[/latex] [latex]2x,7y,-4[/latex]
[latex]3{x}^{2}+4{x}^{2}+5y+3[/latex] [latex]3{x}^{2},4{x}^{2},5y,3[/latex]
Notice that when a term is being subtracted, the coefficient and the term are negative.

example

Identify each term in the expression [latex]9b+15{x}^{2}-a+6[/latex]. Then identify the coefficient of each term.

Solution:

The expression has four terms. They are [latex]9b,15{x}^{2},-a[/latex], and [latex]6[/latex].

  • The coefficient of [latex]9b[/latex] is [latex]9[/latex].
  • The coefficient of [latex]15{x}^{2}[/latex] is [latex]15[/latex].
  • The coefficient of [latex]a[/latex] is [latex]-1[/latex].
  • The coefficient of a constant is the constant, so the coefficient of [latex]6[/latex] is [latex]6[/latex].

try it

What exactly does it mean for a constant and a variable to be multiplied or divided? Let’s consider [latex]4x[/latex]. Multiplication is repeated addition, so [latex]4x=x+x+x+x[/latex].  What about [latex]\frac{1}{5}x[/latex]?  Multiplying by [latex]\frac{1}{5}[/latex] is equivalent to dividing by [latex]5[/latex]. So, [latex]\frac{1}{5}x=\frac{x}{5}[/latex]. In other words, we take the quantity [latex]x[/latex] and divide it by[latex]5[/latex].

Recall that exponents are a more efficient way to write repeated multiplication. So, a variable with an exponent means to multiply the variable the exponent number of times. For example, [latex]x^3=x\cdot x\cdot x[/latex] and [latex]y^5=y\cdot y\cdot y\cdot y\cdot y[/latex].

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

[latex]5x,7,{n}^{2},4,-3x,9{n}^{2}[/latex]

  • The terms [latex]7[/latex] and [latex]4[/latex] are both constant terms.
  • The terms [latex]5x[/latex] and [latex]-3x[/latex] are both terms with [latex]x[/latex].
  • The terms [latex]{n}^{2}[/latex] and [latex]9{n}^{2}[/latex] both have [latex]{n}^{2}[/latex].

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

  • [latex]7[/latex] and [latex]4[/latex] are like terms.
  • [latex]5x[/latex] and [latex]3-x[/latex] are like terms.
  • [latex]{n}^{2}[/latex] and [latex]9{n}^{2}[/latex] are like terms.

Like Terms

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

Like terms are terms where the variables match exactly (exponents included). Examples of like terms would be [latex]5xy[/latex] and [latex]-3xy[/latex], or [latex]8a^2b[/latex] and [latex]a^2b[/latex], or [latex]-\frac{3}{4}[/latex] and [latex]8[/latex].

example

Identify the like terms:

  1. [latex]{y}^{3},7{x}^{2},-14,\frac{2}{3},4{y}^{3},-9x,5{x}^{2}[/latex]
  2. [latex]-4{x}^{2}+2x+5{x}^{2}+6x-40x+8xy[/latex]

Solution:

1. [latex]{y}^{3},7{x}^{2},-14,\frac{2}{3},-4{y}^{3},-9x,5{x}^{2}[/latex]

Look at the variables and exponents. The expression contains [latex]{y}^{3},{x}^{2},x[/latex], and constants.

The terms [latex]{y}^{3}[/latex] and [latex]-4{y}^{3}[/latex] are like terms because they both contain [latex]{y}^{3}[/latex].

The terms [latex]7{x}^{2}[/latex] and [latex]5{x}^{2}[/latex] are like terms because they both contain [latex]{x}^{2}[/latex].

The terms [latex]-14[/latex] and [latex]\frac{2}{3}[/latex] are like terms because they are both constants.

The term [latex]-9x[/latex] does not have any like terms in this list since no other terms have the variable [latex]x[/latex] raised to the power of [latex]1[/latex].

2. [latex]-4{x}^{2}+2x+5{x}^{2}+6x-40x+8xy[/latex]

Look at the variables and exponents. The expression contains [latex]{x}^{2},x,text{and}xy[/latex] terms.

The terms [latex]-4{x}^{2}[/latex] and [latex]5{x}^{2}[/latex] are like terms because they both contain [latex]{x}^{2}[/latex].

The terms [latex]2x,6x,\text{and}-40x[/latex] are like terms because they all contain [latex]x[/latex].

The term [latex]8xy[/latex] has no like terms in the given expression because no other terms contain the two variables [latex]xy[/latex].

 

try it

Simplifying Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. We saw this when adding fractions like [latex]\frac{2}{\color{blue}{7}}+\frac{3}{\color{blue}{7}}=\frac{2+3}{\color{blue}{7}}[/latex]. In words this is [latex]2[/latex] sevenths [latex]+3[/latex] sevenths equals [latex]5[/latex] sevenths. Another example is [latex]12[/latex] oranges [latex]-8[/latex] oranges [latex]=4[/latex] oranges.

If we apply the same reasoning to [latex]3\color{red}{x}+6\color{red}{x}[/latex] we add the coefficients [latex]3+6[/latex] and keep the like term [latex]\color{red}{x}[/latex]:  [latex]3\color{red}{x}+6\color{red}{x}=9\color{red}{x}[/latex]

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 common variable. It doesn’t matter what [latex]x[/latex] is. If we have [latex]3[/latex] of something and add [latex]6[/latex] more of the same thing, the result is [latex]9[/latex] of them.

The expression [latex]3x+6x[/latex] 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, while the Associative Property of Addition says we can regroup the addends without changing the sum. So we could rearrange and group 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.

Combining like terms

  1. Identify like terms.
  2. Rearrange the expression so like terms are together.
  3. Add the coefficients of the like terms and keep the common variable(s).

If we have like terms, we are allowed to add (or subtract) the numbers in front of the variables, then keep the variables the same. Kind of like saying four pens plus three pens equals seven pens. But two pens plus six pencils can’t be combined and simplified because they are not “like terms”. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.

This is shown in the following examples:

example

Simplify the expression: [latex]3x+7+4x+5[/latex].

Solution:

[latex]3x+7+4x+5[/latex]
Identify the like terms. [latex]\color{red}{3x}+\color{blue}{7}+\color{red}{4x}+\color{blue}{5}[/latex]
Rearrange the expression, so the like terms are together. [latex]\color{red}{3x}+\color{red}{4x}+\color{blue}{7}+\color{blue}{5}[/latex]
Add the coefficients of the like terms. .
The original expression is simplified to… [latex]7x+12[/latex]

Example

Combine like terms:  [latex]5x-2y-8x+7y[/latex]

Solution

The like terms in this expression are:

[latex]5x[/latex] and [latex]-8x[/latex]

[latex]-2y[/latex] and [latex]7y[/latex]

Note how we kept the sign in front of each term.

Combine like terms:

[latex]5x-8x = -3x[/latex]

[latex]-2y+7y = 5y[/latex]

Note how signs become operations when we combine like terms.

Simplified Expression:

[latex]5x-2y-8x+7y=-3x+5y[/latex]

try it

Example

Combine like terms:  [latex]x^2-3x+9-5x^2+3x-1[/latex]

Solution

The like terms in this expression are:

[latex]x^2[/latex] and [latex]-5x^2[/latex]

[latex]-3x[/latex] and [latex]3x[/latex]

[latex]9[/latex] and [latex]-1[/latex]

Combine like terms:

[latex]\begin{array}{r}x^2-5x^2 = -4x^2\\-3x+3x=0\,\,\,\,\,\,\,\,\,\,\,\\9-1=8\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Simplified Expression:

[latex]-4x^2+8[/latex]

The video that follows shows another example of combining like terms.  Pay attention to why we are not able to combine all three terms in the example.

example

Simplify the expression: [latex]8x+7{x}^{2}+{x}^{2}+4x[/latex].

Solution:

[latex]8x+7{x}^{2}+{x}^{2}+4x[/latex]
Identify the like terms.  [latex]\color{blue}{8x}+\color{red}{7x^2}+\color{red}{x^2}+\color{blue}{4x}[/latex]
Rearrange the expression so like terms are together.  [latex]\color{red}{7x^2}+\color{red}{x^2}+\color{blue}{8x}+\color{blue}{4x}[/latex]
Add the coefficients of the like terms.  [latex]8x^2+12x[/latex]

These are not like terms and cannot be combined. So [latex]8{x}^{2}+12x[/latex] is in simplest form.

 

try it

The following video presents more examples of how to combine like terms in an algebraic expression.

Multiplication by a Constant and the Distributive Property

Multiplying a term by a constant is the multiplication of the constant and the other elements of the term. For example, [latex]2\cdot 3xy=2\cdot 3\cdot x\cdot y[/latex], which simplifies to [latex]6xy[/latex] through multiplication of the constants. Likewise, [latex]-5\cdot (-7x^2)=(-5\cdot(-7))x^2=35x^2[/latex] and [latex]-\frac{2}{3}\cdot 6x^2y^3=(-\frac{2}{3}\cdot 6)x^2y^3=-4x^2y^3[/latex]. Here we use the Associative Property of Multiplication to regroup the multiplication of the constants.

Examples

Simplify:

1. [latex]5\cdot 4x[/latex]

2. [latex]-2\cdot 7x^2[/latex]

3. [latex]\frac{2}{3}\cdot 5xy[/latex]

4. [latex]-\frac{5}{6}\cdot \frac{4}{5}xy[/latex]

 

Solution

1. [latex]5\cdot 4x=(5\cdot 4)x=20x[/latex]

2. [latex]-2\cdot 7x^2=(-2\cdot 7)x^2=-14x^2[/latex]

3. [latex]\frac{2}{3}\cdot 5xy=\left (\frac{2}{3}\cdot 5\right )xy=\frac{10}{3}xy[/latex]

4. [latex]-\frac{5}{6}\cdot \frac{4}{5}xy=\left (-\frac{5}{6}\cdot \frac{4}{5}\right )xy=-\frac{5\cdot 4}{6\cdot 5}xy=-\frac{5\cdot 2\cdot 2}{2\cdot 3\cdot 5}xy=-\frac{2}{3}xy[/latex]

Try It

Simplify:

1. [latex]8\cdot 3x[/latex]

2. [latex]2\cdot -9x^3[/latex]

3. [latex]\frac{4}{5}\cdot 3xy[/latex]

4. [latex]\frac{4}{7}\cdot \frac{14}{5}x^4[/latex]

 

To multiply an expression that has two or more terms by a constant we have to multiply each term by that constant. This rule is called the Distributive Property of Multiplication over Addition. For example, to multiply the the expression [latex]3xy + 5z[/latex] by [latex]2[/latex] we must distribute the [latex]2[/latex] to both [latex]3xy \text{ and } 5z[/latex].

[latex]\begin{align} &= \color{blue}{2}(3xy + 5z)\\ &=\color{blue}{2}\cdot 3xy +\color{blue}{2}\cdot 5z\\ &=6xy + 10z \end{align}[/latex]

The distributive property of multiplication over addition

For any real number [latex]a[/latex], [latex]a(x + y)=ax+ay[/latex]

Examples

Multiply:

1. [latex]5(4x+7y)[/latex]

2. [latex]-7(5x-y^2)[/latex]

3. [latex]\frac{1}{2}(4x+6xy)[/latex]

4. [latex]-\frac{4}{3}(6x^2+9x-2)[/latex]

5. [latex]-(5x-7)[/latex]

Solution

1. [latex]\color{blue}{5}(4x+7y)=\color{blue}{5}\cdot 4x +\color{blue}{5}\cdot 7y=20x + 35y[/latex]

2. [latex]\color{blue}{-7}(5x-y^2)=\color{blue}{-7}\cdot 5x\color{blue}{-7}\cdot (-y^2)=-35x+7y^2[/latex]

3. [latex]\color{blue}{\frac{1}{2}}(4x-6xy)=\color{blue}{\frac{1}{2}}(4x)-\color{blue}{\frac{1}{2}}(6xy)=2x-3xy[/latex]

4. [latex]\color{blue}{-\frac{4}{3}}(6x^2+9x-2)=\color{blue}{-\frac{4}{3}}(6x^2)\color{blue}{-\frac{4}{3}}(9x)\color{blue}{-\frac{4}{3}}(-2)=-8x^2-12x+\frac{8}{3}[/latex]

5. [latex]\color{blue}{-}(5x-7) =\color{blue}{-1}(5x-7) =\color{blue}{(-1)}5x - \color{blue}{(-1)}7=-5x+7[/latex]

Example 3 shows that fractions can be multiplied onto an expression. Multiplying by [latex]\frac{1}{2}[/latex] is equivalent to dividing by [latex]2[/latex]. This implies that the distributive property also applies to division:

[latex]\begin{align} &= \frac{6x+3}{\color{blue}{3}}\\&= \frac{6x}{\color{blue}{3}}+\frac{3}{\color{blue}{3}}\\ &= 2x + 1\end{align}[/latex]

Examples

Divide:

1. [latex]\frac{5x+30}{5}[/latex]

2. [latex]\frac{11x-44}{11}[/latex]

3. [latex]\frac{8x-40}{-8}[/latex]

4. [latex]\frac{-x+4}{-1}[/latex]

Solution

1. [latex]\frac{5x+30}{\color{blue}{5}}=\frac{5x}{\color{blue}{5}}+\frac{30}{\color{blue}{5}}=x+6[/latex]

2. [latex]\frac{11x-44}{\color{blue}{11}}=\frac{11x}{\color{blue}{11}}-\frac{44}{\color{blue}{11}}=x-4[/latex]

3. [latex]\frac{8x-40}{\color{blue}{-8}}=\frac{8x}{\color{blue}{-8}}-\frac{40}{\color{blue}{-8}}=-x+5[/latex]

4. [latex]\frac{-x+4}{\color{blue}{(-1)}}=\frac{-x}{\color{blue}{(-1)}}+\frac{4}{\color{blue}{(-1)}}=x-4[/latex]

Try It

Divide:

1. [latex]2(2x+40)[/latex]

2. [latex]\frac{15x-45}{5}[/latex]

3. [latex]\frac{12x-40}{-4}[/latex]

4. [latex]\frac{2x-9}{-1}[/latex]

5. [latex]-3(5x-7)[/latex]

 

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