Mathematics • Year 10 • Unit 2 • Lesson 8
Linear Equations in the Real World
Translate real-world setups (ages, ride share, perimeter, mobile plans, swimming) into linear equations, then solve using the balance rule. The "Explain" question asks you to defend your method in words.
1. Word problems
For each problem: (i) define your variable, (ii) write the equation, (iii) solve it, (iv) state the answer in context.
1.1 — Ava and Ben's ages. Ava is three times as old as her brother Ben. In four years, Ava will be twice as old as Ben. Let Ben's current age be b years.
(a) Write expressions for Ava's age now and in 4 years.
(b) Set up an equation that says "Ava in 4 years = 2 × Ben in 4 years".
(c) Solve for b and state Ben's current age. 4 marks
1.2 — Mobile phone plan. A pre-paid plan charges a $15 monthly fee plus $0.12 per call minute. Mai's bill last month was $33.
(a) Write a linear equation for the bill C in terms of minutes m used.
(b) Solve for m to find how many minutes Mai used. 3 marks
1.3 — Triangle perimeter. A triangle has sides x cm, (x + 3) cm and (2x − 1) cm. Its perimeter is 22 cm.
(a) Write an equation by adding the three sides and setting it equal to 22.
(b) Simplify and solve for x.
(c) Find the length of each side. 4 marks
1.4 — Ride share vs taxi. A taxi charges $5 flag fall plus $2.40 per km. A ride share charges $3 booking plus $2.80 per km. Let d be the distance in km.
(a) Write an expression for the cost of each option.
(b) For what value of d do the two options cost the same?
(c) Which is cheaper for a 10 km trip? Show your working. 4 marks
1.5 — Swimming laps. Marco swam 5 laps further than half the number of laps Jess swam. Together they swam 65 laps. Let j = Jess's laps.
(a) Write an expression for Marco's laps in terms of j.
(b) Write an equation for the total and solve for j.
(c) How many laps did Marco swim? 3 marks
2. Explain your thinking
Communication, not just numbers — use full sentences. 4 marks
2.1 A classmate solves 2x + 6 = 14 by dividing both sides by 2 first, getting x + 6 = 7, then subtracting 6 to get x = 1. Their answer is wrong. Using the words "reverse BIDMAS" and "balance rule", explain (i) what mistake the classmate made in step 1, (ii) why it leads to the wrong answer, and (iii) the correct order of operations to solve the equation. Show the correct working.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Ages
(a) Ava now = 3b; in 4 years Ava = 3b + 4 and Ben = b + 4.
(b) 3b + 4 = 2(b + 4).
(c) Expand: 3b + 4 = 2b + 8. Subtract 2b: b + 4 = 8. Subtract 4: b = 4. Ben is 4 years old. (Check: Ava = 12 now; in 4 years Ava = 16 = 2 × 8 = 2 × (Ben + 4) ✓.)
1.2 — Phone plan
(a) C = 0.12m + 15. (b) 33 = 0.12m + 15 → 0.12m = 18 → m = 150 minutes.
1.3 — Triangle perimeter
(a) x + (x + 3) + (2x − 1) = 22. (b) 4x + 2 = 22 → 4x = 20 → x = 5. (c) Sides: 5 cm, 8 cm, 9 cm. Check: 5 + 8 + 9 = 22 ✓.
1.4 — Ride share vs taxi
(a) Taxi = 5 + 2.40d; ride share = 3 + 2.80d. (b) 5 + 2.40d = 3 + 2.80d → 2 = 0.40d → d = 5 km. (c) For 10 km: taxi = 5 + 24 = $29; ride share = 3 + 28 = $31. Taxi is cheaper by $2 at 10 km.
1.5 — Swimming laps
(a) Marco = j/2 + 5. (b) j + (j/2 + 5) = 65 → 1.5j = 60 → j = 40 laps. (c) Marco = 40/2 + 5 = 25 laps. Check: 40 + 25 = 65 ✓.
2.1 — Explain (sample response)
(i) The classmate divided by 2 before dealing with the +6. That breaks reverse BIDMAS: the order of operations applied to x in 2x + 6 is "× 2 then + 6", so to undo we must reverse: subtract 6 first, then divide by 2. (ii) When they divided 2x + 6 by 2 they only divided the 2x and the answer 14, but ignored that the constant 6 must also be divided — they really got x + 3 = 7, not x + 6 = 7 — and even fixing that arithmetic leads to a different value. The balance rule says you can divide both sides by 2, but every term on each side must be divided. (iii) Correct working: 2x + 6 = 14 → subtract 6 → 2x = 8 → divide by 2 → x = 4. Check: 2(4) + 6 = 14 ✓.
Marking: 1 mark for naming the BIDMAS order error; 1 mark for identifying the division-of-every-term issue; 1 mark for correct working; 1 mark for using both required terms in context.