How to solve equations with parentheses

Parentheses tell us which terms belong together. When a number or a minus sign sits outside them, we can distribute it to each term inside. Let’s do that on both sides of an equation and then solve.

Distribute before collecting like terms

Multiply every term inside the parentheses by the factor outside. A minus sign in front acts like multiplying by negative one. Then combine like terms on each side before moving terms across the equation.

Example: Distribute on both sides

- (3 x + 4) = 6 (2 x - 7) + 8

Distribute the negative on the left and the six on the right. Combine the constants on the right.

\begin{array}{l} - 3 x - 4 = 12 x - 42 + 8 \\ - 3 x - 4 = 12 x - 34 \end{array}

Add $3 x$ to both sides, then add $34$ and divide by $15$ .

\begin{array}{l} - 4 = 15 x - 34 \\ 30 = 15 x \\ x = 2 \end{array}

Check $x = 2$ in the original equation. Both sides give $- 10$ .

Example: Simplify powers before distributing

- (b - 2^{0}) = 7^{0} (b^{0} - 4) - 4 (b + 8)

A nonzero base to the zero power equals $1$ . In this example we work with $b \neq 0$ , so $b^{0} = 1$ as well. Then distribute and combine constants.

\begin{array}{l} - (b - 1) = 1 (1 - 4) - 4 (b + 8) \\ - b + 1 = - 35 - 4 b \\ 3 b + 1 = - 35 \\ b = - 12 \end{array}

The solution is nonzero and makes both sides equal to $13$ . If a grouped expression is easier to simplify before distributing, you can do that first. Each step should preserve equality.

Learn mathKrista KingJanuary 21, 2019math, learn online, online course, online math, equations with parentheses, solving simple equations, parentheses in equations

How to solve equations with parentheses

Distribute before collecting like terms

Example: Distribute on both sides

Example: Simplify powers before distributing

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