Solving Two-Step Equations
Solve equations that require two inverse operations — and translate real-world scenarios into two-step equations you can solve.
Most real-world equations aren't one-step problems. Your monthly phone bill has a base fee plus a per-minute charge. A contractor's quote has a flat fee plus hourly labor. A salary formula has a base plus commission. Two-step equations are the bread and butter of practical algebra — once you master this pattern, you can solve 80% of all everyday math problems.
- Identify the two operations applied to the variable and reverse them in the correct order
- Solve two-step equations involving any combination of operations
- Translate word problems into two-step equations and solve them
In a two-step equation like \(3x + 4 = 19\), the variable x has been: (1) multiplied by 3, then (2) had 4 added. To undo: reverse order — undo the addition first, then the multiplication.
\[ 3x + 4 = 19 \xrightarrow{-4} 3x = 15 \xrightarrow{\div 3} x = 5 \]Rule: Undo + and − first (they're "farther" from the variable), then undo × and ÷ (they're "closest" to the variable).
Solve: \(2x - 5 = 11\)
- Add 5 to both sides: \(2x - 5 + 5 = 11 + 5\) → \(2x = 16\)
- Divide both sides by 2: \(x = 8\)
- Check: \(2(8) - 5 = 16 - 5 = 11\) ✓
Solve: \(\dfrac{x}{3} + 7 = 12\)
- Subtract 7 from both sides: \(\dfrac{x}{3} = 5\)
- Multiply both sides by 3: \(x = 15\)
- Check: \(\dfrac{15}{3} + 7 = 5 + 7 = 12\) ✓
A taxi charges a $3 base fare plus $2 per mile. You paid $15. How far did you travel?
- Write the equation: \(2m + 3 = 15\), where m = miles
- Subtract 3: \(2m = 12\)
- Divide by 2: \(m = 6\)
You traveled 6 miles. Check: \(2(6) + 3 = 15\) ✓
- \(4x + 3 = 23\)
- \(\dfrac{n}{2} - 4 = 6\)
- A plumber charges $50 plus $40/hr. The bill was $170. How many hours?
▶ Show Answers
- \(4x = 20\), x = 5.
- \(\frac{n}{2} = 10\), n = 20.
- \(40h + 50 = 170\) → \(40h = 120\) → h = 3 hours.
- Dividing before adding/subtracting: In \(3x + 6 = 21\), don't divide by 3 first. Subtract 6 first, THEN divide. Wrong order gives wrong answer.
- Not applying the operation to both sides: When you subtract 6, subtract from both sides: \(3x + 6 - 6 = 21 - 6\).
- Sign errors when subtracting a negative: \(3x - (-4) = 3x + 4\). Subtracting a negative is addition.
- Undo in reverse PEMDAS order: undo + and − first, then × and ÷.
- Two steps, two inverse operations — apply both to both sides.
- Check every solution by substituting back into the original equation.
- Word problems translate directly: identify the variable, the constant, and the per-unit rate.
Sales professionals constantly solve two-step equations: "If I want to earn $5,000 this month and my base salary is $2,000, how many units do I need to sell at $150 commission each?" That's \(150u + 2000 = 5000\) → \(u = 20\) units. Every commission structure, every tiered pricing model, every bonus threshold is a two-step equation waiting to be solved.
Calculator Connection
Use the Two-Step Equation Solver to verify your solutions — it walks through both inverse operations with the work shown.
Try it with the Calculator
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Solving Two-Step Equations - Quiz
5 questions per attempt · Beginner · Pass at 70%
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