Lesson 23: Solving Two-Step Equations

New Concept

If an equation has two operations, use inverse operations and work backward to undo each operation one at a time.

To reverse the order of operations:

• First add or subtract.
• Then multiply or divide.

What's next

Next, you'll practice solving equations with positive and negative coefficients, work through fraction-based problems, and tackle real-world applications like gym membership costs. We'll also cover common error patterns and verification strategies.

Property

If an equation has two operations, use inverse operations and work backward to undo each operation one at a time. To reverse the order of operations, first add or subtract, and then multiply or divide.

Examples

To solve $3x - 2 = 10$, first add $2$ to both sides to get $3x = 12$. Then, divide by $3$ to find $x = 4$.
To solve $4x + 5 = 17$, first subtract $5$ from both sides to get $4x = 12$. Then, divide by $4$ to find $x = 3$.
To solve $30x + 90 = 1500$, first subtract $90$ to get $30x = 1410$. Then, divide by $30$ to find $x = 47$.

Explanation

Think of it like getting undressed: you untie your shoes before you take them off! To solve an equation, you must reverse the order of operations. First, undo any addition or subtraction to clear out the constants. Then, undo multiplication or division to finally get the variable all by itself.

Property

When you multiply a number by its reciprocal, the product is 1. For any nonzero numbers $a$ and $b$, the reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$, so $\frac{a}{b} \cdot \frac{b}{a} = 1$.

Examples

The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ because $\frac{2}{3} \cdot \frac{3}{2} = \frac{6}{6} = 1$.
The reciprocal of $5$ (which is $\frac{5}{1}$) is $\frac{1}{5}$ because $5 \cdot \frac{1}{5} = 1$.
To solve $\frac{1}{2}n = 8$, multiply both sides by the reciprocal, 2, so $n = 16$.

Explanation

A reciprocal is a number’s “flipping buddy.” When a fraction and its flipped version multiply, they always equal 1. This is a fantastic trick for getting rid of fraction coefficients when solving equations. It's like having a magic wand that turns tricky fractions into the number one, simplifying your problem instantly.

Property

To solve a two-step equation with a negative coefficient, use the same inverse operations to isolate the variable. Remember that dividing by a negative number is the final step to find the solution.

Examples

To solve $8 = -5m + 6$, first subtract $6$ from both sides to get $2 = -5m$. Then divide by $-5$ to find $m = -\frac{2}{5}$.
To solve $-10 = -2x + 12$, first subtract $12$ from both sides to get $-22 = -2x$. Then divide by $-2$ to find $x = 11$.

Explanation

Don't let a negative sign spook you! Just follow the reverse order of operations as always. Undo addition or subtraction first. The most important part is the final step: when you divide to isolate the variable, you must divide by the entire negative coefficient, including its negative sign, to solve correctly.