Simplifying Expressions with Like Terms

Introduction

In this lesson, you will learn about simplifying expressions with like terms. Even if you don’t know what a variable equals, you can sometimes make expressions simpler. This is done by simplifying the expression.

This video illustrates the lesson material below. Watching the video is optional.

Simplifying Expressions with Like Terms (09:10 mins) | Transcript

Combining Like Terms

Here are vocabulary words that will help you understand the lesson better:

Coefficient: The number being multiplied to a variable.
- 2 is the coefficient for 2n
- -3 is the coefficient for -3y
Reduce: Combine or simplify by doing whatever operations you can do.
Term: A part of an expression separated from the rest by an addition or subtraction sign:
- One term: \(3a\), \(6b\), \(7x\)
- Two terms: \(3a + 6b\), \(2x - 5y\)
Like Terms: Any terms in an expression where the variables are the same:
- \(3a\) and \(4a\)
- \(2b^2\) and \(5b^2\)
- Note: \(2b^2\) and \(3b\) are not like terms

Example 1
Simplify this expression: \(3w+4w\)

You can reduce these two terms because they are like terms, meaning they have the same variable (w): \(4w + 3w = 7w\).

Here is another way to look at this:

\begin{align*}
3w &+ 4w\\
(w + w + w) &+ (w + w + w + w)\\
&7w
\end{align*}

This gives the same answer, \(7w\), which is the same as \(7\cdot w\).

Example 2
Simplify this expression: \(2y^3+y^3\)

These variables are able to be combined because they are the same (including the power of three).

\begin{align*} &2y^3+y^3 &\color{red}\small\text{Simplify the expression}\\\\ &2y^3+1y^3 &\color{red}\small\text{Note that \(y^3 = 1y^3\)}\\\\ &3y^3 &\color{red}\small\text{Add the coefficients of like terms} \end{align*}

So \(2y^3+y^3 = 3y^3\).

Example 3
Simplify this expression: \(3w+4y^3-w-y^3\)

You can rearrange the order of the expression into like terms and then combine the like terms using the coefficients of each term. When rearranging the expression, pay careful attention to the operations or signs being used for each term.

\begin{align*}3w+4y^3&-w-y^3 &\color{red}\small\text{Simplify the expression}\\\\3w -w +&4y^3-y^3 &\color{red}\small\text{Rearrange into like terms}\\\\3w -1w +&4y^3-1y^3 &\color{red}\small\text{Note: \(w = 1w\) and \(y^3 = 1y^3\)}\\\\2w +&3y^3 &\color{red}\small\text{Combine like terms}\\\end{align*}

When you combine like terms, you get \(2w+3y^3\). This is as far as you can reduce this expression. You can’t combine these terms because they have different variables.

Order of Operations

Remember to follow the order of operations. Sometimes this means to use the distributive property to solve what’s in the parentheses.

When you see two different letters, you can easily know that you don’t have like terms. But can you add \(3a + 4a^2= ?\) If \(a = 3\), then \(a^2 = 9\). Because these are different numbers, the answer is no, you cannot add \(3a + 4a^2\). Any time you have different letters as variables, or the same letter with different powers, you do not have like terms.

Example 4
Simplify this expression: \(4(2-3b)+2(b-1)\)

Remember the order of operations:

Parentheses
Exponents
Multiplication or division from left to right
Addition or subtraction from left to right

The order of operations tells you to do whatever is in the parentheses first. In this problem you have two sets of parentheses. Look at each and see if you can simplify:

\((2-3b)\): These terms cannot be simplified because 2 and -3b are not like terms.
\((b-1)\): This part of the equation is similar to the one above. You cannot reduce this statement because they are not like terms.

Next, check to see if you have any exponents. Since there aren't any exponents, move on to multiplication and division. There are two examples of multiplication. Begin with the \(4(2-3b)\) by multiplying 4 into everything inside the parentheses:
\begin{align*}
& 4(2 - 3b) &\color{red}\small\text{Simplify the left side}\\\\
& 4(2) - 4(3b) &\color{red}\small\text{Distribute the \(4\) inside the parentheses}\\\\
& 4\times 2 - 4\times 3b &\color{red}\small\text{Multiply from left to right}\\\\
& 8 - 12b &\color{red}\small\text{Left side simplified}
\end{align*}

Since you have simplified this part of the expression, rewrite the equation to look like this: \(8-12b+2(b-1)\). Since you multiplied the content within the parentheses by something outside of the parentheses, the parentheses on that side of the equation will go away.

Move on to the next part of the expression, \(2(b-1)\), and work on multiplying 2 by everything within the parentheses:

\begin{align*}
& 2(b-1) &\color{red}\small\text{Simplify the right side}\\\\
& 2(b) - 2(1) &\color{red}\small\text{Distribute the \(2\) inside the parentheses}\\\\
& 2\times b - 2\times 1 &\color{red}\small\text{Multiply from left to right}\\\\
&2b - 2 &\color{red}\small\text{ Right side simplified}
\end{align*}

Now the expression will read like this: \(8-12b+2b-2\). This can be simplified like Example 3 by rearranging into like terms and then combining like terms.

\begin{align*}8-12b&+2b-2 &\color{red}\small\text{Simplify the expression}\\\\8 -2 - &12b + 2b &\color{red}\small\text{Rearrange into like terms}\\\\6 -&10b &\color{red}\small\text{Combine like terms}\\\end{align*}

This is the answer because \(6\) and \(10b\) are not like terms and cannot be simplified any further. The answer can be written as \(6 - 10b\) or \(-10b + 6\).

Things to Remember

When simplifying expressions, only like terms can be combined.
The order of operations: PEMDAS (parentheses, exponents, multiplication, division, addition, and subtraction).

Practice Problems

\(7w − 2w\) (
Solution

x

Solution: \(5w\)
Details:
The terms \(7w\) and \(2w\) have the same variable, \(w\). They are like terms.

\(7w - 2w\)

The problem seems to have had a w distributed into each term. Using the knowledge of the distributive property, undo the distribution above; this is called factoring. Place the numbers \(7−2\) inside parentheses and the variable, w, outside the parentheses. Like this:

\(w \lgroup 7 - 2\rgroup\)

Subtract \(7-2\).

\(w \lgroup {\color{red}7 - 2}\rgroup\)

Replace the subtraction of \(7−2\) with the answer 5.

\(w \lgroup{\color{Red}5}\rgroup\)

To show the solution in standard mathematical form, remove the parentheses and move the variable to the right of the number.

The simplified expression is \({\color{Red}5}w\).

)
\(5s − 7 − 3s + 11\) (
Solution

x

Solution: \(2s + 4\)
Details:

One way to simplify this problem is to move the like terms, and their signs, near each other.

\(5s − 7 − 3s + 11\)

Move the \(-7\) to the end, next to the \(11\). This allows the two sets of like terms to be placed near each other.

\(5s − 3s + 11 {\color{Red}− 7}\)

Find the difference between \(5s\) and \(3s\).

\({\color{Red}5s − 3s} + 11 − 7\)

Replace \(5s − 3s\) with the difference of \(2s\).

\({\color{Red}2s} + 11 − 7\)

Find the difference between \(11\) and \(7\)

\(2s + {\color{Red}11 − 7}\)

Replace \(11-7\) with the difference of \(4\).

\(2s + {\color{Red}4}\)

This problem simplifies to \(2s + 4\).

)
\(5a − 2b − 6 + 3a + 6b\) (
Video Solution

x

Solution: \(8a + 4b − 6\)
Details:

(Simplifying Expressions with Like Terms #3 (01:48 mins) | Transcript)

| Transcript)
\(2v^{2}+6+3v-3v^{2}\) (
Solution

x

Solution: \(-v^{2}+3v+6\)
Details:
Start by combining the like terms. First find the difference between \(2v^{2}\) and \(-3v^{2}\).

\(2v^{2}+6+3v-3v^{2}\)

The difference between \(2v^{2}\) and \(-3v^{2}\) is \(-1v^{2}\), or \(-v^{2}\). Remove \(2v^{2}\) and \(-3v^{2}\) and replace with \(-v^{2}\).

\(-v^{2}+6+3v\)

Move the terms to standard mathematical form. Move the \(+3v\) in between \(v^{2}\) and \(+6\).

This expression simplifies to \(-v^{2}+3v+6\).

)
\(2 \lgroup 3-2t \rgroup + 5 \lgroup t + 3 \rgroup\) (
Solution

x

Solution:
\(t + 21\)

)
\(\lgroup 4 x + 3 y - 2z \rgroup - 2 \lgroup x + 3 z \rgroup\) (
Video Solution

x

Solution: \(2{\text{x}} + 3{\text{y}} − 8{\text{z}}\)
Details:

(Simplifying Expressions with Like Terms #6 (02:07 mins) | Transcript)

| Transcript)

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