Section 1.2: Solving Multi-Step Equations

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

If $a$, $b$, and $c$ are any numbers, then

If the terms inside parentheses are not like terms, we have no choice but to use the distributive law to simplify the expression.

Examples

• To simplify $5(x+4)$, distribute the 5 to each term: $5(x) + 5(4) = 5x + 20$.
• To simplify $-3(2y-1)$, multiply each inner term by $-3$: $-3(2y) - 3(-1) = -6y + 3$.
• To simplify $(a-5)6$, distribute the 6 from the right: $a(6) - 5(6) = 6a - 30$.

Explanation

The distributive law lets you multiply a number outside parentheses by each term inside. It's like sharing the outside number with every term in the group through multiplication. This is essential when you can't combine the terms inside first.