Adding and subtracting whole expressions uses two things you know: distributing a negative and combining like terms.
Key idea
To add expressions, drop the parentheses and combine like terms. To subtract an expression, subtract every term in it: $-(cx + d) = -cx - d$. Then combine like terms.
Why it works
Subtracting means adding the opposite, as with numbers. The opposite of a whole group is the opposite of each part: $-(x - 2)$ is $-1 \times (x - 2) = -x + 2$. If you took away only the $x$, you would have taken away less than the whole group.
| Step | $(3x + 5) - (x - 2)$ |
|---|---|
| Write the subtraction as adding the opposite | $(3x + 5) + (-x + 2)$ |
| Drop the parentheses | $3x + 5 - x + 2$ |
| Combine like terms | $2x + 7$ |
Worked examples
Worked example: Adding
Simplify $(3x + 5) + (2x - 7)$.
- Drop the parentheses: $3x + 5 + 2x - 7$.
- Combine: $3x + 2x = 5x$ and $5 - 7 = -2$.
Answer: $5x - 2$
Worked example: Subtracting
Simplify $(3x + 5) - (x - 2)$.
- Subtract both terms: $-(x - 2) = -x + 2$.
- $3x + 5 - x + 2 = 2x + 7$.
- Check with $x = 1$: $(3 + 5) - (1 - 2) = 8 - (-1) = 9$, and $2(1) + 7 = 9$ ✓.
Answer: $2x + 7$
Worked example: "Subtract from"
Subtract $(2x - 5)$ from $(7x + 3)$.
- "Subtract $B$ from $A$" means $A - B$: $(7x + 3) - (2x - 5)$.
- $7x + 3 - 2x + 5 = 5x + 8$.
Answer: $5x + 8$
Why does testing one number work as a check? Equivalent expressions are equal for every value of $x$. If your answer and the original give different values for $x = 1$, something went wrong. (Matching for one number is not a full proof, but it catches almost every slip.)
Common mistake
Subtracting only the first term: $(3x + 5) - (x - 2) = 2x + 3$. The $-2$ inside also gets subtracted, which turns it into $+2$. A quick check is to substitute a number, like $x = 1$, into the original and into your answer.