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

Here are vocabulary words 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 \(4a\), \(2b^2\) and \(5b^2\), note that \(2b^2\) 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 \(3a + 4a^2= ?\) Let’s say \(a = 3\), then \(a^2 = 9\). Because these are different numbers, the answer is no, we cannot add \(3a + 4a^2\). 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

1. \(7w − 2w\) (
   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 \(w\lgroup 7 - 2 \rgroup\).
   
   \(w \lgroup 7 - 2\rgroup\)
   
   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 your solution in standard mathematical form, remove the parentheses and move the variable to the right of the number.
   
   The simplified expression is \({\color{Red}5}w\)
   )
2. \(5s − 7 − 3s + 11\) (
   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.
   
   \(5s − 3s + 11 {\color{Red}− 7}\)
   
   Now find the difference between \(5s\) and \(3s\).
   
   \({\color{Red}5s − 3s} + 11 − 7\)
   
   Replace \(5s − 3s\) with the difference of \(2s\).
   
   \({\color{Red}2s} + 11 − 7\)
   
   Now, 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 \(2s + 4\).
   )
3. \(5a − 2b − 6 + 3a + 6b\) (
   Video Solution

   x

   Solution: \(8a + 4b − 6\)
   Details:
   

   
   (Video Source | Transcript)
   )
4. \(2v^{2}+6+3v-3v^{2}\) (
   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 \(-1v^{2}\). 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 \(-v^{2}+3v+6\).
   )
5. \(2 \lgroup 3-2t \rgroup + 5 \lgroup t + 3 \rgroup\) (
   Solution

   x

   Solution:
   \(t + 21\)
   )
6. \(\lgroup 4 x + 3 y - 2z \rgroup - 2 \lgroup x + 3 z \rgroup\) (
   Video Solution

   x

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

   
   (Video Source | Transcript)
   )

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