Lesson 3: Simplifying Expressions by Combining Like Terms

Property

To add like terms: Add the numerical coefficients of the terms. Do not change the variable factors of the terms.
To subtract like terms: Subtract the numerical coefficients of the terms. Do not change the variable factors of the terms.
Replacing an expression by a simpler equivalent one is called simplifying the expression.

Examples

• To simplify $7b + 4b$, we add the coefficients: $(7+4)b = 11b$. We have combined the like terms into a single, simpler term.
• To simplify $12k - (-3k)$, we subtract the coefficients: $(12 - (-3))k = (12+3)k = 15k$. The variable factor $k$ remains unchanged.
• We cannot simplify $8p + 5q$ because $8p$ and $5q$ are not like terms. Their variable factors are different.

Explanation

When combining like terms, you only perform the operation on their coefficients. The variable part just tells you what 'family' of terms you're counting. It's like adding 4 apples and 5 apples to get 9 apples.

Property

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first.
We can rearrange an expression so the like terms are together.
For example, we simplify $3x + 7 + 4x + 5$ by rewriting it as $3x + 4x + 7 + 5$ and then combining like terms to get $7x + 12$.
We were using the Commutative Property of Addition.

Examples

• To simplify $18p + 6q + (-15p) + 5q$, reorder the terms: $18p + (-15p) + 6q + 5q$. This combines to $3p + 11q$.

• To simplify $\frac{7}{15} \cdot \frac{8}{23} \cdot \frac{15}{7}$, reorder the factors to group reciprocals: $\frac{7}{15} \cdot \frac{15}{7} \cdot \frac{8}{23}$. This becomes $1 \cdot \frac{8}{23} = \frac{8}{23}$.

Property

To simplify an expression with multiple variables, group and combine like terms for each variable type independently.

Examples