Solving One-Step Equations

What You’ll Learn

In this lesson you’ll learn how to solve one-step equations by using inverse (opposite) operations to get the variable alone on one side.

A one-step equation has the variable with only one operation applied (plus, minus, multiply, divide).

To solve, do the inverse operation on both sides to keep the equation balanced:

Goal: Isolate the variable (get it by itself on one side, with 1 as coefficient).

Examples:

\begin{aligned} x + 7 &= 12 \\ x + 7 - 7 &= 12 - 7 \\ x &= 5 \end{aligned}

\begin{aligned} 4y &= 20 \\ 4y \div 4 &= 20 \div 4 \\ y &= 5 \end{aligned}

\begin{aligned} z - 3 &= 9 \\ z - 3 + 3 &= 9 + 3 \\ z &= 12 \end{aligned}

\begin{aligned} w \div 6 &= 8 \\ w \div 6 \times 6 &= 8 \times 6 \\ w &= 48 \end{aligned}

Always do the same thing to both sides. That’s what keeps it fair.

Solve each:

1. $x + 9 = 15$

\begin{aligned} x + 9 &= 15 \\ x + 9 - 9 &= 15 - 9 \\ x &= 6 \end{aligned}

2. $3n = 24$

\begin{aligned} 3n &= 24 \\ 3n \div 3 &= 24 \div 3 \\ n &= 8 \end{aligned}

3. $m - 5 = 11$

\begin{aligned} m - 5 &= 11 \\ m - 5 + 5 &= 11 + 5 \\ m &= 16 \end{aligned}

Check: Plug back in to verify (e.g., for $x = 6$ : $6 + 9 = 15$ . Correct).

One-step equations solve everyday “unknown” problems:

You have some money x and add 25 dollars, now have 70 dollars:

\begin{aligned} x + 25 &= 70 \\ x &= 45 \text{ dollars} \end{aligned}

A tank holds 120 gallons. You use some amount and have 45 left:

\begin{aligned} 120 - y &= 45 \\ y &= 75 \text{ gallons used} \end{aligned}

Pay rate is 15 dollars per hour. You earned 120 dollars:

\begin{aligned} 15h &= 120 \\ h &= 8 \text{ hours worked} \end{aligned}

These quick solves help track spending, budgets, or amounts in recipes/work.

Solve $x + 12 = 35$.

Solve $7y = 56$.

Solve $m - 18 = 42$.

You earned 15 dollars per hour and made 120 dollars total. How many hours did you work?

Solve $\frac{n}{4} = 9$.