Solving Multi-Step Equations

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Solve linear equations that require multiple steps.
  • Use the distributive property and combine like terms before isolating the variable.
  • Recognize when to simplify each side first.

Key Ideas

Multi-step equations often involve:

  • parentheses (distribution)
  • combining like terms
  • constants or variables on both sides

General plan:

  1. Simplify each side (distribute, combine like terms).
  2. Move variable terms to one side and constants to the other.
  3. Isolate the variable using one-step or two-step methods.
  4. Check your solution by substitution.

Common Problem Types

1. Distribute, Then Solve

Solve:
\[ 2(x + 3) = 14 \]

Distribute: \(2x + 6 = 14\)
Subtract 6: \(2x = 8\)
Divide: \(x = 4\)

2. Combine Like Terms

Solve:
\[ 3x + 2x - 5 = 10 \]

Combine: \(5x - 5 = 10\)
Add 5: \(5x = 15\)
Divide: \(x = 3\)

3. Parentheses on One Side

Solve:
\[ 4(x - 2) + 3 = 15 \]

Distribute: \(4x - 8 + 3 = 15\)
Combine: \(4x - 5 = 15\)
Add 5: \(4x = 20\)
Divide: \(x = 5\)

Strategies

  • Simplify first, then move terms.
  • Write work in separate, clear steps.
  • Keep track of variable terms vs. constants.
  • Rewrite the equation neatly after large steps to avoid errors.

Worked Examples

Example 1

Solve:
\[ 5(x + 1) - 2 = 3x + 6 \]

Step-by-step Solution:

  1. Distribute → \(5x + 5 - 2 = 3x + 6\)
  2. Combine → \(5x + 3 = 3x + 6\)
  3. Subtract \(3x\) → \(2x + 3 = 6\)
  4. Subtract 3 → \(2x = 3\)
  5. Divide → \(x = \tfrac{3}{2}\)

Answer: \(\tfrac{3}{2}\)

Example 2

Solve:
\[ 7 - 3(x - 2) = 1 \]

Step-by-step Solution:

  1. Distribute → \(7 - 3x + 6 = 1\)
  2. Combine → \(13 - 3x = 1\)
  3. Subtract 13 → \(-3x = -12\)
  4. Divide by \(-3\) → \(x = 4\)

Answer: \(4\)

WarningCommon Mistakes
  • Forgetting to distribute a negative sign: \(-(x - 3)\) becomes \(-x + 3\).
  • Combining unlike terms (e.g., \(x\) with \(x^2\)).
  • Skipping steps and losing track of signs.
  • Moving terms before simplifying each side.

Practice Problems

  1. \(3(x + 4) = 21\)
  2. \(2x + 5x - 4 = 17\)
  3. \(6(x - 1) - 2 = 10\)
  4. \(10 - 2(x + 3) = 0\)
  5. \(4(x + 2) + 3x = 3\)

1. \(3(x + 4) = 21\)
Distribute → \(3x + 12 = 21\)
Subtract 12 → \(3x = 9\)
Divide → \(x = 3\)

2. \(2x + 5x - 4 = 17\)
Combine → \(7x - 4 = 17\)
Add 4 → \(7x = 21\)
Divide → \(x = 3\)

3. \(6(x - 1) - 2 = 10\)
Distribute → \(6x - 6 - 2 = 10\)
Combine → \(6x - 8 = 10\)
Add 8 → \(6x = 18\)
Divide → \(x = 3\)

4. \(10 - 2(x + 3) = 0\)
Distribute → \(10 - 2x - 6 = 0\)
Combine → \(4 - 2x = 0\)
Subtract 4 → \(-2x = -4\)
Divide → \(x = 2\)

5. \(4(x + 2) + 3x = 3\)
Distribute → \(4x + 8 + 3x = 3\)
Combine → \(7x + 8 = 3\)
Subtract 8 → \(7x = -5\)
Divide → \(x = -\tfrac{5}{7}\)

Summary

  • Simplify each side before moving terms.
  • Use distribution and combining like terms to clean up the equation.
  • Move variables to one side and constants to the other.
  • Check your solution to confirm correctness.
  • Always simplify both sides before moving anything across the equals sign.
  • Remember: distributing negatives is the #1 source of errors—be careful.
  • Combine like terms early to reduce clutter.
  • Neatly rewriting the equation after major steps keeps mistakes away.