# Solving Multi-Step Linear Equations

We sometimes need more than two operations to solve a linear equation . Use inverse operations to undo each operation in reverse order.

To solve the equation $x-5x+7=39$ , we need more than two operations.

By the Identity Property, $x=1x$ .

$1x-5x+7=39$

Combine the like terms.

$-4x+7=39$

Undo the addition. Subtract $7$ from each side.

$-4x+7-7=39-7$

Simplify.

$-4x=32$

Undo the multiplication. Divide each side by $-4$ .

$\frac{-4x}{-4}=\frac{32}{-4}$

Simplify.

$x=-8$

We have solved the equation.

Example:

Solve $2\left(4m+5\right)=3m-20$ . Check the solution.

Solution

Use the distributive law on the left side of the equation.

$\begin{array}{l}2\left(4m\right)+2\left(5\right)=3m-20\\ 8m+10=3m-20\end{array}$

Subtract $3m$ from each side and simplify.

$\begin{array}{l}8m-3m+10=3m-3m-20\\ 5m+10=-20\end{array}$

Undo the addition. Subtract $10$ from each side.

$\begin{array}{l}5m+10-10=-20-10\\ 5m=-30\end{array}$

Undo the multiplication. Divide each side by $5$ .

$\frac{5m}{5}=\frac{-30}{5}$

Simplify.

$m=-6$

To check the solution, substitute $-6$ for $m$ in the equation.

$2\left(4\left(-6\right)+5\right)\stackrel{?}{=}3\left(-6\right)-20$

Simplify.

$\begin{array}{l}2\left(-24+5\right)\stackrel{?}{=}-18-20\\ 2\left(-19\right)\stackrel{?}{=}-18-20\\ -38\stackrel{?}{=}-38\text{\hspace{0.17em}}\end{array}$