The Order of Operations and Variables:

Simplifying Expressions with Like Terms

Even without knowing what a variable is, we can sometimes make expressions with variables look simpler. This is done by simplifying our expression.

Here is a vocabulary word that will help you understand the lesson better:

Coefficient = the number being multiplied to a variable (in 2n, 2 is the coefficient)
Reduce = combine or simplify by doing whatever operations we can
Term = a part of an expression separated from the rest by addition (in 3a + 6b, 3a is one term and 6b is another term)
Like Terms = any terms in an expression where the variables are the same (3a and 4a, \(2{\text{b}}^{2}\) and \(5{\text{b}}^{2}\), note that \(2{\text{b}}^{2}\) and 3b are not like terms)

Video Source (09:10 mins) | Transcript

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

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

Additional Resources

Khan Academy: Intro to Combining Like Terms (04:32 mins, Transcript)
Khan Academy: Simplifying Expressions (04:06 mins, Transcript)
Khan Academy: Combining Like Terms - Challenge Problem (04:38 mins, Transcript)

Practice Problems

Simplify the following expressions:

7w − 2w

5s − 7 − 3s + 11

5a − 2b − 6 + 3a + 6b

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

\( 2(3-2{\text{t}}) + 5 ({\text{t}} + 3) \)

\( ( 4 {\text{x}} + 3 {\text{y}} - 2{\text{z}} ) - 2 ( {\text{x}} + 3 {\text{z}}) \)

Solutions

5w (Written Solution)

The terms 7w and 2w have the same variable, w. They are like terms.

\(7{\text{w}} {-} 2{\text{w}}\)

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 \({\text{w}} (7 - 2) \).

\({\text{w}} (7 - 2)\)

Subtract \(7-2\).

\(\text w({\color{Red} 7-2})\)

Replace the subtraction of 7−2 with the answer 5.

\(\text w({\color{Red} 5})\)

To show your solution in standard mathematical form, remove the parentheses and move the variable to the right of the number. The simplified expression is 5w.

\(5\)\({\text{w}}\)

2s + 4 (Written Solution)

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 − 7

Now find the difference between 5s and 3s.

5s − 3s + 11 − 7

Replace 5s − 3s with the difference of 2s.

2s + 11 − 7

Now, find the difference between 11 and 7

2s + 11 − 7

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

2s + 4

This problem simplifies to 2s + 4

8a + 4b − 6 (Solution Video | Transcript)

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

Start by combining the like terms. First find the difference between \(2{\text{v}}^{2}\) and \(-3{\text{v}}^{2}\).

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

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

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

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

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

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

t + 21

2x + 3y − 8z (Solution Video | Transcript)