Math Core

Some equations come wrapped in parentheses, like $5(2n - 3) = 4n + 21$ . You can't collect like terms while some of them are locked inside a group. The distributive property unlocks them, and then the equation turns into one you already know how to solve.

Distribute first, then solve

Recall that $a(b + c) = ab + ac$ . The number outside multiplies every term inside. Once the parentheses are gone, you can combine like terms on each side and solve just like before.

A plan for any linear equation

Distribute to clear the parentheses.
Combine like terms on each side.
Collect the variable terms on one side and the constants on the other.
Divide (or multiply) to get the variable alone.
Check in the original equation.

Worked example: One set of parentheses

Solve $5(2n - 3) = 4n + 21$ .

\begin{aligned} 5(2n - 3) &= 4n + 21 \\ 10n - 15 &= 4n + 21 && \text{distribute} \\ 6n - 15 &= 21 && \text{subtract } 4n \\ 6n &= 36 && \text{add } 15 \\ n &= 6 && \text{divide by } 6 \end{aligned}

Check: $5(2 \cdot 6 - 3) = 5(9) = 45$ and $4(6) + 21 = 45$ . ✓

Distributing a negative

When the number in front is negative, it multiplies every term inside, and it changes each sign. A minus sign in front of parentheses is the same as multiplying by $-1$ .

Worked example: A negative in front

Solve $-2(x - 5) = 3x - 10$ .

Multiply each term inside by $-2$ : $(-2)(x) = -2x$ and $(-2)(-5) = +10$ .

\begin{aligned} -2(x - 5) &= 3x - 10 \\ -2x + 10 &= 3x - 10 \\ 10 &= 5x - 10 && \text{add } 2x \\ 20 &= 5x \\ 4 &= x \end{aligned}

Check: $-2(4 - 5) = -2(-1) = 2$ and $3(4) - 10 = 2$ . ✓

Common mistake

Distribute to every term, and watch the signs. The two most common errors are:

Multiplying only the first term: $-2(x - 5) \ne -2x - 5$ .
Forgetting that subtracting a group flips every sign inside: $12 - (a - 8) = 12 - a + 8$ , not $12 - a - 8$ .

Parentheses on both sides

Sometimes both sides need distributing. Do each side separately, then solve.

Worked example: Distribute on both sides

Solve $4(y - 2) = 2(y + 5)$ .

\begin{aligned} 4(y - 2) &= 2(y + 5) \\ 4y - 8 &= 2y + 10 \\ 2y - 8 &= 10 \\ 2y &= 18 \\ y &= 9 \end{aligned}

Check: $4(9 - 2) = 28$ and $2(9 + 5) = 28$ . ✓

Worked example: Distribute, then combine like terms

Solve $2(3a + 1) - (a - 8) = 35$ .

Distribute the $2$ , and distribute the invisible $-1$ in front of the second group.

\begin{aligned} 2(3a + 1) - (a - 8) &= 35 \\ 6a + 2 - a + 8 &= 35 \\ 5a + 10 &= 35 && \text{combine like terms} \\ 5a &= 25 \\ a &= 5 \end{aligned}

Check: $2(16) - (-3) = 32 + 3 = 35$ . ✓

A shortcut: divide first

If the number outside the parentheses divides evenly into the other side, you can divide first. For $3(x + 4) = 27$ , divide both sides by $3$ to get $x + 4 = 9$ , so $x = 5$ . Distributing gives the same answer: $3x + 12 = 27$ , so $3x = 15$ and $x = 5$ . Use whichever way feels cleaner.

Tip

Fractions in front of parentheses often cancel nicely. In $\dfrac{1}{3}(6x - 9) = x + 5$ , distributing gives $2x - 3 = x + 5$ , so $x = 8$ . Check: $\dfrac{1}{3}(48 - 9) = \dfrac{1}{3}(39) = 13$ and $8 + 5 = 13$ . ✓