Equations with the Distributive Property

When an equation has parentheses, like $3(x + 4) = 27$ , the first job is to get rid of them. The distributive property does that: multiply the number outside the parentheses by each term inside.

After you distribute, the parentheses are gone and what's left is a two-step equation you already know how to solve. Distribute first, then solve — that's the whole lesson.

Distribute first

$3(x + 4)$ means three copies of the whole quantity $x + 4$ . The distributive property multiplies the $3$ by each term inside: $3 \cdot x + 3 \cdot 4 = 3x + 12$ . So the equation $3(x + 4) = 27$ becomes $3x + 12 = 27$ .

The most common mistake is multiplying only the first term — writing $3x + 4$ instead of $3x + 12$ . The outside number multiplies every term inside the parentheses, not just the one it touches.

Distributing a negative

If the outside number is negative, its sign travels with the multiplication. $-4(x - 2)$ becomes $-4x + 8$ : the $-4$ times $x$ gives $-4x$ , and the $-4$ times $-2$ gives $+8$ . Both terms change, and a negative times a negative comes out positive.

A shortcut: divide both sides first

When the outside number divides evenly into the other side, you can divide both sides by it instead of distributing. In $3(x + 4) = 27$ , dividing both sides by $3$ gives $x + 4 = 9$ , so $x = 5$ .

Use whichever route you like — distributing always works, and the shortcut is nice when the division is clean.

Worked examples

Example 1: distribute a positive number

Solve $3(x + 4) = 27$ .

Distribute the

3

to both terms

3x + 12 = 27

Subtract

12

from both sides

3x = 15

Divide both sides by

3

x = 5

Check:

3(5 + 4) = 3(9) = 27

✓

Answer: $x = 5$

Example 2: a two-term inside expression

Solve $2(3x - 5) = 26$ .

Distribute the

2

to both terms

6x - 10 = 26

Add

10

to both sides

6x = 36

Divide both sides by

6

x = 6

Answer: $x = 6$

Example 3: distribute a negative number

Solve $-4(x - 2) = 20$ .

Distribute the

-4

— both signs change

-4x + 8 = 20

Subtract

8

from both sides

-4x = 12

Divide both sides by

-4

x = -3

Answer: $x = -3$

Try one yourself

Common questions

Do I always have to distribute first?

No. If the outside number divides evenly into the other side, dividing both sides by it is faster: $3(x + 4) = 27$ becomes $x + 4 = 9$ in one move. But distributing works on every problem, so it's the safe default.

What happens when I distribute a negative number?

Every term inside the parentheses picks up the sign. $-4(x - 2)$ is $-4x + 8$ , not $-4x - 8$ — the $-4$ times $-2$ is a positive $8$ . Sign errors here are the number one source of wrong answers in this topic.

What if there's more outside the parentheses, like

2(x + 3) + 5 = 19

Distribute first, then combine like terms: $2x + 6 + 5 = 19$ becomes $2x + 11 = 19$ , so $2x = 8$ and $x = 4$ . Clearing the parentheses is always the opening move.

How do I check my answer?

Substitute it into the original equation, parentheses and all. For $x = -3$ : $-4(-3 - 2) = -4(-5) = 20$ is true, so the answer checks.

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Equations with the Distributive Property

Distribute first

Distributing a negative

A shortcut: divide both sides first

Worked examples

Example 1: distribute a positive number

Example 2: a two-term inside expression

Example 3: distribute a negative number

Try one yourself

Common questions

Keep learning

Two-Step Equations

Variables on Both Sides

One-Step Equations (Multiply / Divide)

The Distributive Property