Mathematics • Year 10 • Unit 2 • Lesson 12
Word Problems with Linear Equations — Skill Drill
Build fluency with the five-step process from Lesson 12: read, define, write, solve, check. One worked example, one guided trace, then eight independent problems covering translation, consecutive numbers, money and distance.
1. I do — fully worked example
Translate, solve, then check. Every step has a reason underneath so you can see why, not just what.
Problem. The sum of two consecutive even integers is 54. Find the integers.
Step 1 — Read carefully and define the variable.
Let n = the first even integer. Then the next consecutive even integer is n + 2.
Reason: consecutive even (or odd) integers are 2 apart, not 1. Express both unknowns in ONE variable.
Step 2 — Write the equation.
n + (n + 2) = 54
Reason: the sum of the two integers equals 54.
Step 3 — Solve.
2n + 2 = 54 → 2n = 52 → n = 26
Reason: combine like terms, subtract 2, divide by 2.
Step 4 — State the full answer and check against the original sentence.
The two integers are 26 and 28. Check: 26 + 28 = 54 ✓, both are even ✓.
Reason: always state ALL the numbers (not just n = 26) and check against the words, not just the equation.
Answer: The integers are 26 and 28.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 5 marks
Problem. A taxi charges a $5 flag fall plus $2.50 per kilometre. A fare costs $32.50. How many kilometres were travelled?
Step 1 — Define the variable: Let d = ________________________ in kilometres.
Step 2 — Write the cost equation:
fixed flag fall + (per-km rate × distance) = total fare
____ + ____ d = ____
Step 3 — Solve:
2.5 d = ____ − ____ = ____ (subtract the flag fall from both sides)
d = ____ ÷ ____ = ____ km (divide by the per-km rate)
Step 4 — Check against the original sentence:
Total = $5 + $2.50 × ____ = $____ + $____ = $____. Matches $32.50? ____ (yes/no).
Final answer: The taxi travelled ________ km.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single skill). The middle two are standard (combine two tools). The last two are extension.
Foundation — single skill
3.1 Translate "five more than twice a number is 17" into an equation, then solve for the number. 1 mark
3.2 A number is doubled and then 7 is subtracted. The result is 15. Write and solve an equation to find the original number. 1 mark
3.3 A car travels at 80 km/h for 2.5 hours. Use d = rt to calculate the distance. 1 mark
3.4 Three consecutive integers sum to 48. Define the variable, write the equation, then state the largest of the three. 1 mark
Standard — combine tools
3.5 The sum of three consecutive odd integers is 81. Define n, write the equation, solve, then state all three integers. 2 marks
3.6 A rectangle has perimeter 36 cm and its length is 4 cm more than its width. Let w = width. Write an equation for the perimeter and solve for w. 2 marks
Extension — push your thinking
3.7 Adult cinema tickets cost $22 and child tickets cost $14. A group of 10 people pays $172 in total. Define a and c, write two equations, then solve for the number of adults and children. 3 marks
3.8 A train leaves Sydney at 9:00 am at 90 km/h. A car leaves Sydney on the same route at 10:00 am at 120 km/h. Let t = hours after 10:00 am. Write an equation for when the car catches the train, solve for t, then state the catch-up time. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded taxi problem)
Step 1: d = distance travelled in km.
Step 2: 5 + 2.5 d = 32.5.
Step 3: 2.5 d = 32.5 − 5 = 27.5; d = 27.5 ÷ 2.5 = 11 km.
Step 4: Total = $5 + $2.50 × 11 = $5 + $27.50 = $32.50. Matches? Yes.
Final answer: 11 km.
3.1 — "Five more than twice a number is 17"
Let n be the number. Equation: 2n + 5 = 17. Subtract 5: 2n = 12. Divide by 2: n = 6. Check: 2(6) + 5 = 17 ✓.
3.2 — Doubled, then subtract 7, gives 15
Equation: 2n − 7 = 15. Add 7: 2n = 22. Divide by 2: n = 11. Check: 2(11) − 7 = 15 ✓.
3.3 — Car at 80 km/h for 2.5 hours
d = rt = 80 × 2.5 = 200 km.
3.4 — Three consecutive integers sum to 48
Let n = first. Equation: n + (n+1) + (n+2) = 48 → 3n + 3 = 48 → n = 15. Integers: 15, 16, 17. Largest = 17. Check: 15 + 16 + 17 = 48 ✓.
3.5 — Three consecutive odd integers sum to 81
Let n = first odd integer. Equation: n + (n+2) + (n+4) = 81 → 3n + 6 = 81 → 3n = 75 → n = 25. Integers: 25, 27, 29. Check: 25 + 27 + 29 = 81 ✓.
3.6 — Rectangle perimeter 36 cm, length = width + 4
Let w = width, length = w + 4. Perimeter = 2w + 2(w + 4) = 4w + 8 = 36. So 4w = 28, w = 7 cm. Length = 11 cm. Check: 2(7) + 2(11) = 14 + 22 = 36 ✓.
3.7 — Cinema tickets, 10 people, $172 total
Let a = adults, c = children. Equations: a + c = 10 and 22a + 14c = 172.
From the first: c = 10 − a. Substitute: 22a + 14(10 − a) = 172 → 22a + 140 − 14a = 172 → 8a = 32 → a = 4. Then c = 10 − 4 = 6.
Check: 4 + 6 = 10 ✓; 22(4) + 14(6) = 88 + 84 = 172 ✓. The group has 4 adults and 6 children.
3.8 — Train at 9 am 90 km/h, car at 10 am 120 km/h
Head start: train travels 90 km in 1 hour. After 10 am, train distance = 90 + 90t, car distance = 120t. Equation: 120t = 90 + 90t. Subtract 90t: 30t = 90 → t = 3 hours. Catch-up time = 10:00 am + 3 hours = 1:00 pm.