Introduction

Two-step equations are a natural extension of one-step equations.
They involve two operations applied to the variable, such as:

• Multiply then add
• Divide then subtract
• Add then multiply
• Subtract then divide

Your goal is to undo both operations using inverse operations, one step at a time.

This article assumes you already understand how to solve one-step equations.

What Are Two-Step Equations?

A two-step equation is an equation where the variable has:

• A coefficient (a number multiplying it), and
• A constant term added or subtracted.

Examples:

• $3x + 5 = 17$
• $\dfrac{x}{4} - 2 = 6$
• $7 - 2x = 1$

Each requires two inverse operations to isolate $x$.

General Strategy

To solve a two-step equation:

1. Undo addition or subtraction first.
   • This removes the constant term.
2. Undo multiplication or division second.
   • This isolates the variable.

Think of it as peeling away layers:

• Outer layer: $+5$, $-3$, $+7$, etc.
• Inner layer: $\times 4$, $\div 2$, etc.

Example

Solve $3x + 5 = 17$:

• Subtract $5$ from both sides: $$3x = 12$$
• Divide both sides by $3$: $$x = 4$$

Worked Examples

Example 1: $4x - 3 = 13$

• Add $3$ to both sides: $4x = 16$
• Divide by $4$: $x = 4$

Example 2: $\dfrac{x}{5} + 6 = 10$

• Subtract $6$: $\dfrac{x}{5} = 4$
• Multiply by $5$: $x = 20$

Example 3: $7 - 2x = 1$

• Subtract $7$: $-2x = -6$
• Divide by $-2$: $x = 3$

Example 4: $3(x - 2) = 12$

• Divide by $3$: $x - 2 = 4$
• Add $2$: $x = 6$

Common Mistakes to Avoid

• Forgetting to apply the inverse operation to *both* sides.
• Changing the order of steps.
  Always remove the constant term before the coefficient.
• Sign errors, especially with subtraction or negative coefficients.
• Not distributing when needed, e.g., $2(x + 3)$.

Calculator

Exercises

1. Solve the equation: $5x + 7 = 32$

   Solution

   $5x + 7 = 32$
   Subtract $7$: $5x = 25$
   Divide by $5$: $x = 5$
2. Solve: $\dfrac{x}{3} - 4 = 2$

   Solution

   $\dfrac{x}{3} - 4 = 2$
   Add $4$: $\dfrac{x}{3} = 6$
   Multiply by $3$: $x = 18$
3. Solve: $9 - 2x = 1$

   Solution

   $9 - 2x = 1$
   Subtract $9$: $-2x = -8$
   Divide by $-2$: $x = 4$
4. Solve: $4(x - 1) = 20$

   Solution

   $4(x - 1) = 20$
   Divide by $4$: $x - 1 = 5$
   Add $1$: $x = 6$
5. Solve: $\dfrac{2x + 6}{4} = 5$

   Solution

   $\dfrac{2x + 6}{4} = 5$
   Multiply by $4$: $2x + 6 = 20$
   Subtract $6$: $2x = 14$
   Divide by $2$: $x = 7$
6. Solve: $3x - 8 = -2$

   Solution

   $3x - 8 = -2$
   Add $8$: $3x = 6$
   Divide by $3$: $x = 2$
7. Solve: $7 + \dfrac{x}{2} = 10$

   Solution

   $7 + \dfrac{x}{2} = 10$
   Subtract $7$: $\dfrac{x}{2} = 3$
   Multiply by $2$: $x = 6$
8. Solve: $6 - 3(x + 1) = 0$

   Solution

   $6 - 3(x + 1) = 0$
   Distribute: $6 - 3x - 3 = 0$
   Simplify: $3 - 3x = 0$
   Subtract $3$: $-3x = -3$
   Divide by $-3$: $x = 1$