Topic Review — Linear Models and Further Algebra — Solutions

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1. Simplifying and solving equations

   1. 3x + 7 = 22: x = 5
   2. 5y − 4 = 16: y = 4
   3. 2(m + 3) = 14: m + 3 = 7, m = 4
   4. 4n − 9 = n + 6: 3n = 15, n = 5
   5. 3a + 5b − a + 2b: 2a + 7b
   6. 4x × 3y: 12xy
   7. (t + 1) / 3 = 5: t + 1 = 15, t = 14
   8. 2(3k − 4) = 16: 3k − 4 = 8, k = 4

2. Describing patterns algebraically

   1. Tile rule: T = 2n + 1
   2. Row 10: T = 2(10) + 1 = 21 tiles
   3. Sequence 4, 9, 14, 19: T = 5n − 1
   4. T = 3n + 2, 8th term: T = 3(8) + 2 = 26
   5. T = 5n − 3 = 47: 5n = 50, n = 10 (10th term)
   6. Is 30 in T = 2n + 6?: 2n + 6 = 30, 2n = 24, n = 12. Yes, 30 is the 12th term.

3. Creating and interpreting linear models

   1. Phone plan equation: C = 15 + 0.10m
   2. Cost for 120 minutes: C = 15 + 0.10(120) = $27
   3. Bill of $32: 32 = 15 + 0.10m, 0.10m = 17, m = 170 minutes
   4. Gradient and intercept: Gradient = 0.10 (cost increases $0.10 per minute); y-intercept = 15 (base monthly fee with no calls)
   5. Taxi equation: F = 3.50 + 2.20k
   6. Alex vs Bella: Alex: F = 3.50 + 2.20(8) = $21.10. Bella: F = 3.50 + 2.20(5) + 4 = $18.50. Alex pays more.

4. Interpreting and comparing linear models

   1. Plumber equations: Plumber A: C = 80h + 50. Plumber B: C = 60h + 90.
   2. Equal cost: 80h + 50 = 60h + 90, 20h = 40, h = 2 hours
   3. 3-hour job: A: 80(3)+50=$290. B: 60(3)+90=$270. Plumber B is cheaper for 3 hours.
   4. Intersection at (4, 370): At 4 hours, both tradespeople charge $370. Below 4 hours one is cheaper; above 4 hours the other is cheaper.

5. Problem solving with linear models and algebra

   1. Gym membership:
      1. A: C = 200 + 25m. B: C = 45m. Equal when 200 + 25m = 45m, 20m = 200, m = 10 months.
      2. After 12 months: A = 200 + 25(12) = $500; B = 45(12) = $540. Option A is better for 12 months.
   2. Matchstick pattern:
      1. M = 3n + 1
      2. M = 3(20) + 1 = 61 matchsticks
      3. 3n + 1 = 100, 3n = 99, n = 33. The 33rd shape uses 100 matchsticks.
   3. Draining container: V = 240 − 12t. Empty when V = 0: t = 20 minutes. Half full when V = 120: 120 = 240 − 12t, 12t = 120, t = 10 minutes.