Mathematics • Year 7 • Unit 2 • Lesson 14
Two-Step Equations in the Real World
Turn "fixed fee plus per-item cost" situations — taxi fares, gym memberships, party budgets — into two-step equations, then solve.
1. Word problems
For each: choose a letter, write a two-step equation, solve using SADMEP (undo +/− first, then ×/÷), and answer the question in a sentence. Always check.
1.1 — Taxi fare. A taxi charges a $5 flag fall plus $2 per kilometre. A trip cost $27 in total.
(a) Let d be the distance in km. Write an equation.
(b) Solve for d and write the answer in a sentence. 2 marks
1.2 — Gym membership. A gym charges a one-off $30 sign-up fee plus $15 per month. After m months, Ben has paid $135 in total.
(a) Write an equation in m.
(b) Solve for m and check by substitution. 2 marks
1.3 — Party balloons. A party shop charges a flat $8 delivery fee plus $1.50 per balloon. The total bill for the party was $50.
(a) Let b be the number of balloons ordered. Write an equation.
(b) Solve for b. 2 marks
1.4 — Class score. A class quiz gives 4 marks per correct answer and starts everyone with a 2-mark bonus. Sara scored 30 marks in total.
(a) Let c be the number of correct answers. Write an equation.
(b) Solve for c and answer in a sentence. 2 marks
1.5 — Drained pool. A pool is being drained at a steady rate. Each hour, 200 L is pumped out. The pool started with 1700 L of water and now has only 500 L remaining.
(a) Let h be the number of hours the pump has been running. Write an equation in the form 1700 − 200h = 500.
(b) Solve for h and answer in a sentence. 2 marks
2. Explain your thinking
This question is about communication, not just symbols. Use full sentences. 4 marks
2.1 A classmate solving 3x + 7 = 19 says: "First I'll divide by 3 because BEDMAS does multiplication before addition." In your own words, explain (i) why their order is wrong, (ii) what SADMEP means and why we use it when solving, (iii) what the correct first step is and what equation results, and (iv) what the final solution is and how to check it.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Taxi fare
(a) Equation: 2d + 5 = 27.
(b) Subtract 5: 2d = 22. Divide by 2: d = 11 km. The trip was 11 km long. Check: 2(11) + 5 = 27 ✓.
1.2 — Gym membership
(a) Equation: 15m + 30 = 135.
(b) Subtract 30: 15m = 105. Divide by 15: m = 7 months. Check: 15(7) + 30 = 105 + 30 = 135 ✓.
1.3 — Party balloons
(a) Equation: 1.5b + 8 = 50.
(b) Subtract 8: 1.5b = 42. Divide by 1.5: b = 28 balloons. Check: 1.5(28) + 8 = 42 + 8 = 50 ✓.
1.4 — Class score
(a) Equation: 4c + 2 = 30.
(b) Subtract 2: 4c = 28. Divide by 4: c = 7 correct answers. Sara got 7 questions right. Check: 4(7) + 2 = 28 + 2 = 30 ✓.
1.5 — Drained pool
(a) Equation: 1700 − 200h = 500.
(b) Subtract 1700: −200h = −1200. Divide by −200: h = 6 hours. The pump has been running for 6 hours. Check: 1700 − 200(6) = 1700 − 1200 = 500 ✓.
2.1 — Explain your thinking (sample response)
(i) Dividing by 3 first would give (3x + 7) ÷ 3 = 19 ÷ 3, which produces x + 7⁄3 = 19⁄3 — uglier than what we started with, and the +7 hasn't been dealt with cleanly. BEDMAS tells us the order operations were applied; to UNDO them we go in the OPPOSITE order.
(ii) SADMEP stands for Subtract / Add, then Divide / Multiply, then Exponents, then Parentheses — the reverse of BEDMAS. When solving, the LAST operation applied is the FIRST we undo.
(iii) The correct first step is subtract 7 from both sides, giving the cleaner equation 3x = 12.
(iv) Then divide both sides by 3 to get x = 4. Check: 3(4) + 7 = 12 + 7 = 19 ✓.
Marking: 1 for spotting that dividing first creates fractions; 1 for SADMEP / reverse-of-BEDMAS; 1 for the correct first step; 1 for the final answer + check.