Simplify expressions using combining like terms and the distributive property. | Linear equations in one variable | SAT Math

Distribute first, then combine like terms to simplify an expression before solving.

Core Idea

Before solving, simplify each side: distribute to remove parentheses, then combine terms that share the same variable (or are both constants).

Understanding

The distributive property says $𝑎 (𝑏 + 𝑐) = 𝑎 𝑏 + 𝑎 𝑐$ . On the SAT, you'll use this to clear parentheses before doing anything else. Pay close attention to distributing negative signs — $- (3 𝑥 - 2)$ becomes $- 3 𝑥 + 2$ , not $- 3 𝑥 - 2$ .

Like terms share the same variable raised to the same power. So $5 𝑥$ and $- 2 𝑥$ are like terms (combine to $3 𝑥$ ), but $5 𝑥$ and $5 𝑥 2$ are not. Constants like $7$ and $- 3$ are also like terms.

Simplifying first often cuts a problem in half. An equation like $2 (𝑥 + 4) + 3 𝑥 = 28$ looks like it has many moving parts, but after distributing and combining, it's just $5 𝑥 + 8 = 28$ — a basic two-step equation.

Step by Step

Apply the distributive property to remove all parentheses.
Identify like terms on each side of the equation.
Combine like terms (add or subtract their coefficients).
Proceed to solve the simplified equation.

Misconceptions

Dropping the sign when distributing a negative: $- 2 (𝑥 - 5)$ is $- 2 𝑥 + 10$ , not $- 2 𝑥 - 10$ .
Combining unlike terms — for example, adding $3 𝑥 + 4$ to get $7 𝑥$ instead of leaving them separate.
Only distributing to the first term inside the parentheses and ignoring the rest.

Question

Worked Example

Which expression is equivalent to $3 (2 𝑥 - 4) + 5 𝑥 - 1$ ?

Select an answer to see the explanation