Training Linear Equations and Inequalities Multi-Step Equations
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Multi-Step Equations

22 min Linear Equations and Inequalities

Multi-Step Equations

Multi-step equations require three or more operations to solve. They may involve distributing, combining like terms, or moving variable terms from one side to the other. The strategy, however, remains the same — simplify each side first, then use inverse operations to isolate the variable.

This lesson walks you through the systematic process: distribute any parentheses, combine like terms on each side, collect all variable terms on one side, and then solve the resulting one- or two-step equation.

Mastering multi-step equations is critical because nearly every word problem and real-world mathematical model produces equations of this type.

These may involve the distributive property, combining like terms, or variables on both sides.

Strategy
  1. Distribute to remove parentheses.
  2. Combine like terms on each side.
  3. Move variable terms to one side.
  4. Move constants to the other side.
  5. Divide by the coefficient.
Example 1

$2(3x - 4) + 5 = 19$

  1. Distribute: $6x - 8 + 5 = 19$
  2. Combine: $6x - 3 = 19$
  3. Add 3: $6x = 22$
  4. Divide: $x = \dfrac{22}{6} = \dfrac{11}{3}$

Variables on Both Sides

Example 2

$5x - 3 = 2x + 9$

  1. Subtract $2x$: $3x - 3 = 9$
  2. Add 3: $3x = 12$
  3. Divide: $x = 4$

Check: $5(4) - 3 = 17$ and $2(4) + 9 = 17$ ✓

Example 3

$4(x - 3) = 2(x + 5)$

  1. Distribute: $4x - 12 = 2x + 10$
  2. Subtract $2x$: $2x - 12 = 10$
  3. Add 12: $2x = 22$
  4. $x = 11$
Example 4

$\dfrac{2x + 1}{3} = 5$

Multiply by 3: $2x + 1 = 15$. Then $2x = 14$, $x = 7$.

Special Cases

Watch For
  • No solution (contradiction): $2x + 1 = 2x + 5$ → $1 = 5$ (false)
  • All real numbers (identity): $3(x + 2) = 3x + 6$ → $0 = 0$ (always true)

Practice Problems

1. $3(x + 4) = 27$
2. $7x - 3 = 4x + 9$
3. $2(x - 5) + 3x = 15$
4. $5(2x + 1) = 3(x + 4)$
5. $\dfrac{x - 3}{4} = 2$
6. $6x + 4 = 6(x + 1)$ — classify
7. $3(x - 2) = 3x - 6$ — classify
8. $8 - 2(3x + 1) = -4x$
9. $\dfrac{3x}{2} + 4 = 13$
10. $4(x - 1) - 2(x + 3) = 0$
11. $0.3x + 1.2 = 0.5x - 0.4$
12. $\dfrac{x}{2} + \dfrac{x}{3} = 10$
13. $-2(4x - 1) = 3(x - 4) + 3$
14. $\dfrac{2(x+3)}{5} = 4$
15. $5x - (2x + 7) = 3(x - 1)$ — classify
Show Answer Key

1. $x = 5$

2. $x = 4$

3. $x = 5$

4. $x = 1$

5. $x = 11$

6. No solution ($4 \neq 6$)

7. Identity (all real numbers)

8. $x = 3$

9. $x = 6$

10. $x = 5$

11. $x = 8$

12. $x = 12$

13. $x = 1$

14. $x = 7$

15. No solution ($-7 \neq -3$)

Solving One- and Two-Step Equations Solving Linear Inequalities