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Solving Multi-Step Equations

Solve equations with variables on both sides, parentheses, and multiple steps — including translating complex word problems into equations.

Lesson 7 of 11 Algebra & Equations Intermediate ⏱ 10 min read
🔥 Why This Matters

Real formulas are rarely two-step problems. Mortgage amortization, physics equations, break-even analyses, and engineering formulas all require multiple steps: distributing terms, collecting variables from both sides, and simplifying before solving. This lesson is the bridge from basic algebra to the math used in professional work.

🎯 What You'll Learn
  • Distribute multiplication over parentheses before solving
  • Collect variable terms from both sides of an equation
  • Solve equations with 3 or more steps systematically
📖 Key Vocabulary
Distributive Property\(a(b+c) = ab + ac\) — multiply the outside factor by each term inside the parentheses. Variables on Both SidesWhen the variable appears on both the left and right — collect to one side first. CollectMove all variable terms to one side and all constants to the other.
Key Concept — Multi-Step Strategy

Follow this order every time:

  1. Distribute to clear all parentheses
  2. Combine like terms on each side
  3. Collect variables to one side (add/subtract the variable term)
  4. Isolate the variable with inverse operations (undo + −, then × ÷)
Worked Example 1 — Distribute then Solve

Solve: \(3(x + 4) = 24\)

  1. Distribute: \(3x + 12 = 24\)
  2. Subtract 12: \(3x = 12\)
  3. Divide by 3: \(x = 4\)
  4. Check: \(3(4+4) = 3(8) = 24\) ✓
Worked Example 2 — Variables on Both Sides

Solve: \(5x - 3 = 2x + 9\)

  1. Subtract 2x from both sides: \(3x - 3 = 9\)
  2. Add 3 to both sides: \(3x = 12\)
  3. Divide by 3: \(x = 4\)
  4. Check: \(5(4)-3 = 17\) and \(2(4)+9 = 17\) ✓
\[ x = 4 \]
Worked Example 3 — Real World: Break-Even Analysis

Company A charges $200 setup + $15/item. Company B charges $50 setup + $25/item. At what quantity do both companies cost the same?

  1. Set costs equal: \(15n + 200 = 25n + 50\)
  2. Subtract 15n: \(200 = 10n + 50\)
  3. Subtract 50: \(150 = 10n\)
  4. Divide by 10: \(n = 15\)

At 15 items, both companies cost the same ($425). Above 15, Company A is cheaper.

✏️ Quick Check
  1. \(2(3x - 1) + 4 = 16\)
  2. \(7x + 5 = 3x + 21\)
  3. Two car rentals: Rent-A costs $30/day + $0.20/mile; EasyRent costs $20/day + $0.30/mile. How many miles until EasyRent is more expensive?
▶ Show Answers
  1. Distribute: \(6x - 2 + 4 = 16\) → \(6x + 2 = 16\) → \(6x = 14\) → x = 7/3 ≈ 2.33
  2. \(4x = 16\) → x = 4
  3. \(0.20m + 30 = 0.30m + 20\) → \(10 = 0.10m\) → m = 100 miles
⚠️ Common Mistakes
  • Distributing to only the first term: \(3(x+4) \neq 3x + 4\). You must multiply 3 by EVERY term inside: \(3x + 12\).
  • Moving the wrong variable term: Subtract the smaller variable coefficient. If you have \(5x\) on left and \(2x\) on right, subtract \(2x\) from both sides — don't subtract \(5x\) and get a negative variable.
  • Forgetting to distribute the negative sign: \(-(x + 3) = -x - 3\), not \(-x + 3\).
✅ Key Takeaways
  • Order matters: Distribute → Combine like terms → Collect variables → Isolate.
  • Move variables to one side before isolating — pick the side that avoids negatives.
  • Distribute carefully — multiply the factor by every term inside the parentheses.
  • Check your answer in the original equation — this catches every error.
💼 Career Connection — Business & Operations

Operations managers use multi-step equations to find break-even points, optimize staffing, and compare vendor pricing. Data analysts write formulas with multiple terms that must be simplified before they can be evaluated. Any role that involves comparing two cost models or optimizing a variable is applying multi-step algebra every day.

Calculator Connection

The Multi-Step Equation Solver handles equations with variables on both sides and parentheses — it shows every step including distribution and collection.

Try it with the Calculator

Apply what you've learned with this tool.

Multi-Step Equation Solver
Solve linear equations with variables on both sides of the equation.
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