Two-Step Equations
Two-step equations like 3x + 7 = 22 appear in 85% of middle school algebra assessments, yet students consistently struggle with the order of operations needed to isolate variables. These foundational skills directly connect to CCSS 7.EE and 8.EE standards, forming the backbone for advanced algebraic thinking.
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Why it matters
Two-step equations model countless real-world scenarios students encounter daily. A cell phone plan charges $25 monthly plus $0.10 per text becomes 25 + 0.10x = 35 when solving for allowable texts on a $35 budget. Store pricing follows similar patterns: if a shirt costs $15 plus 8% tax, the equation 1.08x = 32.40 determines the pre-tax price. Construction workers use 2x + 12 = 50 to find lumber lengths when accounting for waste. Financial literacy emerges through savings problems where 4x + 200 = 800 calculates monthly deposits needed to reach goals. These applications demonstrate why mastering two-step equations at the 7th-grade level creates mathematical confidence that extends far beyond the classroom into practical decision-making skills students will use throughout their lives.
How to solve two-step equations
Two-Step Equations
- Undo the addition/subtraction first (isolate the term with x).
- Then undo the multiplication/division.
- Verify by substituting back.
Example: 3x + 5 = 20 → 3x = 15 → x = 5.
Worked examples
Solve for x: 2x + 4 = 12
Answer: x = 4
- Identify the goal → 2x + 4 = 12 — solve for x — We want to get x alone on one side. This takes two steps: first remove the constant, then remove the coefficient.
- Step 1: Subtract 4 from both sides → 2x + 4 − 4 = 12 − 4 → 2x = 8 — Undo the addition/subtraction to isolate the term with x.
- Step 2: Divide both sides by 2 → 2x ÷ 2 = 8 ÷ 2 → x = 4 — Undo the multiplication. 8 ÷ 2 = 4.
- Verify by substituting back → 2·(4) + 4 = 8 + 4 = 12 ✓ — Replace x with our answer in the original equation. Both sides should be equal.
A student solved 3x + 8 = 14 like this: Step 1: 3x = 14 + 8 = 22 Step 2: x = 22 ÷ 3 = 7 Find and correct the error.
Answer: x = 2
- Identify the error → Step 1 is wrong: should subtract 8, not add it — To undo + 8, we subtract 8 from both sides.
- Correct Step 1 → 3x = 14 − 8 = 6 — Subtract the constant correctly.
- Correct Step 2 → x = 6 ÷ 3 = 2 — Divide to find x.
- Verify → 3·(2) + 8 = 6 + 8 = 14 ✓ — Substitute back to confirm.
Tom is 3 times as old as Sara plus 4 years. Together they are 24 years old. How old is Sara?
Answer: Sara = 5
- Define variable → Let Sara's age = x, Tom's age = 3x + 4 — Express Tom's age in terms of Sara's.
- Write equation → x + (3x + 4) = 24 → 4x + 4 = 24 — Their ages sum to the total.
- Subtract 4 from both sides → 4x = 24 − 4 = 20 — Isolate the x term.
- Divide both sides by 4 → x = 20 ÷ 4 = 5 — Sara is 5 years old.
- Verify → Tom = 3×5+4 = 19, 5+19 = 24 ✓ — Ages add up correctly.
Common mistakes
- ✗Students often add the constant instead of subtracting it. For 2x + 5 = 13, they write 2x = 13 + 5 = 18, giving x = 9 instead of the correct x = 4.
- ✗Many students divide by the constant instead of the coefficient. In 3x + 6 = 21, they calculate x = 21 ÷ 6 = 3.5 rather than first getting 3x = 15, then x = 5.
- ✗Students frequently forget to apply operations to both sides. For 4x - 7 = 25, they write 4x = 25 instead of 4x = 32, leading to x = 6.25 rather than x = 8.
- ✗When dealing with subtraction, students add instead of subtracting. For 5x - 9 = 16, they compute 5x = 16 - 9 = 7, giving x = 1.4 instead of x = 5.
Practice on your own
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