Introduction

Multi-step equations appear when solving requires more than one inverse operation, and often when variables appear on both sides of the equation.
Because you already understand Two-Step Equations, this article builds on that knowledge and introduces:

• Combining like terms
• Distributing
• Moving variable terms to one side
• Keeping equations balanced through multiple steps

The goal is to help you confidently solve equations such as:

• $3x + 4 = 2x - 5$
• $5(x - 2) = 3x + 6$
• $\frac{x + 4}{3} = 2x - 1$

What Are Multi-Step Equations?

A multi-step equation is any equation that requires three or more operations to isolate the variable.

Common features include:

• Variables on both sides
• Parentheses requiring distribution
• Fractions requiring clearing denominators
• Like terms that must be combined

Examples:

• $4x - 7 = 2x + 5$
• $3(x + 2) - 4 = 2(x - 1)$
• $\frac{2x - 3}{5} + 1 = x$

General Strategy

To solve multi-step equations, follow this reliable sequence:

1. Remove parentheses
   • Use distribution: $a(b + c) = ab + ac$
2. Clear fractions (optional but helpful)
   • Multiply both sides by the least common denominator (LCD)
3. Combine like terms
   • On each side separately
4. Move variable terms to one side
   • Add/subtract to collect all $x$ terms together
5. Move constants to the other side
   • Add/subtract numbers
6. Isolate the variable
   • Divide or multiply as needed
7. Check your solution
   • Substitute back into the original equation

Worked Examples

Example 1: $3x + 4 = 2x - 5$

• Subtract $2x$ from both sides
  → $x + 4 = -5$
• Subtract $4$
  → $x = -9$

Example 2: $5(x - 2) = 3x + 6$

• Distribute
  → $5x - 10 = 3x + 6$
• Subtract $3x$
  → $2x - 10 = 6$
• Add $10$
  → $2x = 16$
• Divide by $2$
  → $x = 8$

Example 3: $\frac{x + 4}{3} = 2x - 1$

• Multiply both sides by $3$
  → $x + 4 = 6x - 3$
• Subtract $x$
  → $4 = 5x - 3$
• Add $3$
  → $7 = 5x$
• Divide
  → $x = \frac{7}{5}$

Common Mistakes to Avoid

• Forgetting to distribute a negative sign
• Moving constants and variable terms in the wrong direction
• Not multiplying every term when clearing fractions
• Combining unlike terms (e.g., $3x + 4$ cannot become $7x$)

Calculator

Exercises

1. Solve the equation: $4x - 3 = 2x + 9$

   Solution

   $4x - 3 = 2x + 9$

   • Subtract $2x$: $2x - 3 = 9$
   • Add $3$: $2x = 12$
   • Divide: $x = 6$
2. Solve: $5(x + 1) = 3x + 11$

   Solution

   $5(x + 1) = 3x + 11$

   • Distribute: $5x + 5 = 3x + 11$
   • Subtract $3x$: $2x + 5 = 11$
   • Subtract $5$: $2x = 6$
   • Divide: $x = 3$
3. Solve: $\frac{2x - 5}{3} = x + 1$

   Solution

   $\frac{2x - 5}{3} = x + 1$

   • Multiply by $3$: $2x - 5 = 3x + 3$
   • Subtract $2x$: $-5 = x + 3$
   • Subtract $3$: $x = -8$
4. Solve: $7x + 4 = 3(2x - 1)$

   Solution

   $7x + 4 = 3(2x - 1)$

   • Distribute: $7x + 4 = 6x - 3$
   • Subtract $6x$: $x + 4 = -3$
   • Subtract $4$: $x = -7$
5. Solve: $6 - 2(x - 3) = 4x + 8$

   Solution

   $6 - 2(x - 3) = 4x + 8$

   • Distribute: $6 - 2x + 6 = 4x + 8$
   • Combine: $12 - 2x = 4x + 8$
   • Add $2x$: $12 = 6x + 8$
   • Subtract $8$: $4 = 6x$
   • Divide: $x = \frac{2}{3}$
6. Solve: $\frac{x + 2}{4} + 3 = x$

   Solution

   $\frac{x + 2}{4} + 3 = x$

   • Multiply by $4$: $x + 2 + 12 = 4x$
   • Combine: $x + 14 = 4x$
   • Subtract $x$: $14 = 3x$
   • Divide: $x = \frac{14}{3}$
7. Solve: $9x - 5 = 2(4x + 1)$

   Solution

   $9x - 5 = 2(4x + 1)$

   • Distribute: $9x - 5 = 8x + 2$
   • Subtract $8x$: $x - 5 = 2$
   • Add $5$: $x = 7$
8. Solve: $3(x - 4) + 2 = x - 10$

   Solution

   $3(x - 4) + 2 = x - 10$

   • Distribute: $3x - 12 + 2 = x - 10$
   • Combine: $3x - 10 = x - 10$
   • Subtract $x$: $2x - 10 = -10$
   • Add $10$: $2x = 0$
   • Divide: $x = 0$