Mathematics Standard • Year 11 • Module 1 • Lesson 2
Solving One-Step and Two-Step Equations
Practise HSC-style writing on equations — three multi-mark short answers and one structured extended response with marking criteria.
1. Short-answer questions
1.1 Solve 6x = 54 and check your solution by substitution. 2 marks Band 3
1.2 Solve 4x + 9 = 37 and check your solution by substitution. 3 marks Band 3-4
1.3 A removalist company charges a $90 callout fee plus $55 per hour. The final bill came to $337.50.
(a) Let h be the number of hours billed. Write an equation that models this bill.
(b) Solve the equation for h.
(c) State, in a sentence, how many hours the job took. 4 marks Band 4
2. Extended response
2.1 A community pool offers two membership options.
Option A: No joining fee, $9.50 per visit.
Option B: $40 joining fee, $6.00 per visit.
(a) Let v be the number of visits. Write a cost equation for each option.
(b) Solve the equation that finds when both options cost the same.
(c) Calculate the total cost at that break-even number of visits, and verify both options give the same amount.
(d) Explain in a clear conclusion sentence which option is cheaper for fewer than the break-even number of visits, and which is cheaper for more. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — both cost equations correctly written: A: C = 9.50v; B: C = 40 + 6.00v.
Part (b) — 3 marks
• 1 mark — sets the two expressions equal: 9.50v = 40 + 6.00v.
• 1 mark — subtracts 6.00v from both sides: 3.50v = 40.
• 1 mark — divides by 3.50: v ≈ 11.43, or states "12 visits is the first whole-number value where Option B is cheaper".
Part (c) — 1 mark
• 1 mark — substitutes v = 40/3.50 into both formulas and shows both give the same cost (≈ $108.57).
Part (d) — 2 marks
• 1 mark — names the cheaper option for FEWER than ~11.43 visits (Option A).
• 1 mark — names the cheaper option for MORE than ~11.43 visits (Option B), in a clear sentence.
Your response:
Stuck? Set the two cost expressions equal: 9.50v = 40 + 6.00v, then collect like terms. The decimal answer tells you it's actually 12 visits before Option B becomes the cheaper one.How did this worksheet feel?
What I'll revisit before next class:
1.1 — 6x = 54 (2 marks)
Sample response. Divide both sides by 6: x = 54/6 = 9. Check: 6(9) = 54 ✓.
Marking notes. 1 mark — correct value. 1 mark — substitution check shown. A bare "x = 9" with no check scores 1/2.
1.2 — 4x + 9 = 37 (3 marks)
Sample response.
Subtract 9 from both sides: 4x = 28.
Divide both sides by 4: x = 7.
Check: 4(7) + 9 = 28 + 9 = 37 ✓.
Marking notes. 1 mark — subtracts 9 from both sides correctly. 1 mark — divides by 4 to get x = 7. 1 mark — substitution check shown. Common error: dividing by 4 first gives the wrong sequence (x + 9/4 = 37/4) and usually scores 0.
1.3 — Removalist bill (4 marks)
(a) Sample response. 90 + 55h = 337.50.
(b) Sample response. Subtract 90: 55h = 247.50. Divide by 55: h = 4.5.
(c) Sample response. The job took 4.5 hours (i.e. 4 hours 30 minutes) of work.
Marking notes. 1 mark — correct equation. 1 mark — correct subtraction of 90. 1 mark — correct division by 55 giving 4.5. 1 mark — final sentence with units in hours (or hours and minutes). A bare "h = 4.5" with no concluding sentence scores 3/4.
2.1 — Pool memberships (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Cost equations.
Option A: C = 9.50v. Option B: C = 40 + 6.00v. [1 mark — both correct.]
(b) Break-even equation.
Set the costs equal: 9.50v = 40 + 6.00v. [1 mark — equation set up.]
Subtract 6.00v from both sides: 3.50v = 40. [1 mark — collects like terms.]
Divide both sides by 3.50: v = 40 ÷ 3.50 = 11.43 (to 2 d.p.). [1 mark — correct solution.]
(c) Check at the break-even.
Substitute v = 40/3.50 = 11.4286 into each formula:
Option A: C = 9.50 × 11.4286 ≈ $108.57.
Option B: C = 40 + 6.00 × 11.4286 = 40 + 68.57 ≈ $108.57. ✓ [1 mark — both sides verified equal.]
(d) Which is cheaper.
For fewer than 11.43 visits, Option A (pay-per-visit, no joining fee) is cheaper. [1 mark — names A for low usage.]
Conclusion: For 12 or more visits per quarter, Option B is cheaper because the $3.50/visit saving covers the $40 joining fee. For 11 or fewer visits, Option A is cheaper because there is no joining cost to recover. [1 mark — clear conclusion with both ranges.]
Total: 7/7.
Band descriptors for marker.
Band 3: Writes one cost equation correctly; does not set them equal. ≈ 2 marks.
Band 4: Both equations and the set-up equation correct; minor slip in algebra or no conclusion. ≈ 4 marks.
Band 5: Full algebra correct including v ≈ 11.43; conclusion identifies the break-even but does not state which option wins on each side. ≈ 5-6 marks.
Band 6: Complete, both equations correct, v ≈ 11.43 found, verification shown, AND a conclusion sentence naming the cheaper option for both fewer and more visits. 7/7.