Finding the Rule for a Linear Relationship — Solutions

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1. Find the rule (y = mx + c) from tables of values

   1. x: 0,1,2,3 | y: 2,5,8,11: m = 3, c = 2 → y = 3x + 2
   2. x: 0,1,2,3 | y: 4,2,0,−2: m = −2, c = 4 → y = −2x + 4
   3. x: 0,1,2,3 | y: −1,3,7,11: m = 4, c = −1 → y = 4x − 1
   4. x: 0,2,4,6 | y: 1,5,9,13: diff in y = 4, diff in x = 2 → m = 2, c = 1 → y = 2x + 1
   5. x: −1,0,1,2 | y: −4,−1,2,5: m = 3, c = −1 (when x=0, y=−1) → y = 3x − 1
   6. x: 0,1,2,3 | y: 10,7,4,1: m = −3, c = 10 → y = −3x + 10
   7. x: 0,1,2,3 | y: 0,6,12,18: m = 6, c = 0 → y = 6x
   8. x: 1,2,3,4 | y: 4,7,10,13: m = 3; when x=1, y=4 → 4 = 3(1)+c → c = 1 → y = 3x + 1

2. Write the rule given gradient and y-intercept

   1. m = 4, c = −3: y = 4x − 3
   2. m = −2, c = 5: y = −2x + 5
   3. m = 1, c = 0: y = x
   4. m = −5, c = −1: y = −5x − 1
   5. m = 3, c = 7: y = 3x + 7
   6. m = ½, c = 4: y = ½x + 4
   7. m = −1, c = 8: y = −x + 8
   8. m = 0, c = −6: y = −6

3. Find the rule from two points

   1. (0, 3) and (2, 7): m = (7−3)÷(2−0) = 2, c = 3 → y = 2x + 3
   2. (0, −1) and (3, 5): m = (5−(−1))÷3 = 2, c = −1 → y = 2x − 1
   3. (1, 4) and (3, 10): m = (10−4)÷(3−1) = 3; 4 = 3(1)+c → c = 1 → y = 3x + 1
   4. (2, 5) and (5, 11): m = (11−5)÷(5−2) = 2; 5 = 2(2)+c → c = 1 → y = 2x + 1
   5. (0, 8) and (4, 0): m = (0−8)÷4 = −2, c = 8 → y = −2x + 8
   6. (−1, 6) and (3, −2): m = (−2−6)÷(3−(−1)) = −2; 6 = −2(−1)+c → c = 4 → y = −2x + 4
   7. (2, 3) and (6, 7): m = (7−3)÷(6−2) = 1; 3 = 1(2)+c → c = 1 → y = x + 1
   8. (1, −5) and (4, 4): m = (4−(−5))÷(4−1) = 3; −5 = 3(1)+c → c = −8 → y = 3x − 8

4. Verbal descriptions

   1. “Starts at 6, decreases by 2”: y = −2x + 6
   2. “Passes through origin, gradient 5”: y = 5x
   3. “y-intercept −3, gradient 4”: y = 4x − 3
   4. “Horizontal line, y-axis at −2”: y = −2
   5. “Gradient −1, passes through (3, 0)”: 0 = −1(3)+c → c = 3 → y = −x + 3
   6. “x-axis at 4, gradient 2”: 0 = 2(4)+c → c = −8 → y = 2x − 8

5. Same or different rule?

   1. Table A: y=2x+3; Table B: m=2, when x=3, y=9 → y=2x+3: Same rule: y = 2x + 3
   2. Table A: m=3, c=1 → y=3x+1; Table B: m=3, c=1 → y=3x+1: Same rule: y = 3x + 1
   3. Table A: m=4, c=−2 → y=4x−2; Table B: m=4, check: 10=4(3)−2=10 ✓: Same rule: y = 4x − 2
   4. Table A: m=−1, c=5 → y=−x+5; Table B: m=−1, x=1: −1+c=4 → c=5 → y=−x+5: Same rule: y = −x + 5

6. Real-world linear rules

   1. Phone plan: m = (3.50−2.25)÷(10−5) = $0.25/min; c = 2.25−0.25(5) = $1.00 → C = 0.25m + 1.00
   2. Taxi: m = (15.80−10.20)÷(8−4) = $1.40/km; flag fall = 10.20−1.40(4) = $4.60 → C = 1.40d + 4.60
   3. Zara’s savings: m = (125−85)÷(7−3) = $10/week; starting savings = 85−10(3) = $55 → S = 10w + 55
   4. Missing values: m = (9−(−3))÷(6−0) = 2, c = −3; y = 2x−3. When y=1: x=2. When x=4: y=5. Table: x: 0,2,4,6 / y: −3,1,5,9