Solving Two-Step Equations

What You’ll Learn

In this lesson you’ll learn how to solve two-step equations by using two inverse operations in the correct order to get the variable alone.

A two-step equation has two operations applied to the variable (e.g., multiply then add, or subtract then divide).

To solve:

Undo the addition/subtraction first (because it’s attached to the variable term).
Then undo the multiplication/division (to get the coefficient to 1).

Always do the same operation to both sides. Follow reverse order of operations (undo PEMDAS backwards: add/subtract before multiply/divide).

Examples:

\begin{aligned} 3x + 5 &= 20 \\ 3x &= 15 \\ x &= 5 \end{aligned}

\begin{aligned} 4y - 12 &= 8 \\ 4y &= 20 \\ y &= 5 \end{aligned}

\begin{aligned} 2(z + 7) &= 18 \\ z + 7 &= 9 \\ z &= 2 \end{aligned}

Check your solution by substituting back into the original equation.

Solve $5x - 8 = 17$

Step 1 - Add 8 to both sides (undo subtraction):

\begin{aligned} 5x - 8 + 8 &= 17 + 8 \\ 5x &= 25 \end{aligned}

Step 2 - Divide both sides by 5 (undo multiplication):

\begin{aligned} 5x \div 5 &= 25 \div 5 \\ x &= 5 \end{aligned}

Check: $5(5) - 8 = 25 - 8 = 17$ ✓

Another: Solve $2y + 9 = 3$

\begin{aligned} 2y + 9 &= 3 \\ 2y &= 3 - 9 = {-}6 \\ y &= {-}6 \div 2 = {-}3 \end{aligned}

Check: $2(-3) + 9 = -6 + 9 = 3$ ✓

Two-step equations solve many everyday “unknown” problems with multiple parts:

You pay a 15-dollar flat fee plus 2 dollars per mile for a ride. Total cost is 35 dollars. How many miles?

\begin{aligned} 15 + 2m &= 35 \\ 2m &= 20 \\ m &= 10 \text{ miles} \end{aligned}

A savings account starts with some amount x. You add 50 dollars and then double it to reach 180 dollars:

\begin{aligned} 2(x + 50) &= 180 \\ x + 50 &= 90 \\ x &= 40 \text{ dollars} \end{aligned}

These are common in budgeting, travel, work rates, or splitting costs.

Solve $4x + 7 = 23$.

Solve $3y - 12 = 9$.

A parking fee is 5 dollars base plus 3 dollars per hour. You paid 20 dollars. How many hours?

Solve $2(z - 4) = 10$.

Solve $\frac{x}{3} + 5 = 11$.