Finding Equations of Lines — Solutions

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1. Equation from two points

   1. (1,5) and (3,9): m = (9−5)/(3−1) = 2; 5 = 2(1)+b ⇒ b = 3. y = 2x + 3
   2. (0,2) and (4,10): m = (10−2)/(4−0) = 2; b = 2 (y-intercept). y = 2x + 2
   3. (2,7) and (6,3): m = (3−7)/(6−2) = −1; 7 = −1(2)+b ⇒ b = 9. y = −x + 9
   4. (−1,1) and (3,9): m = (9−1)/(3−(−1)) = 2; 1 = 2(−1)+b ⇒ b = 3. y = 2x + 3
   5. (0,−3) and (5,7): m = (7−(−3))/(5−0) = 2; b = −3. y = 2x − 3
   6. (−2,6) and (2,−2): m = (−2−6)/(2−(−2)) = −2; 6 = −2(−2)+b ⇒ b = 2. y = −2x + 2
   7. (1,−4) and (4,5): m = (5−(−4))/(4−1) = 3; −4 = 3(1)+b ⇒ b = −7. y = 3x − 7
   8. (3,3) and (7,7): m = (7−3)/(7−3) = 1; 3 = 1(3)+b ⇒ b = 0. y = x

2. Equation from gradient and point

   1. m=3, (1,4): 4 = 3(1)+b ⇒ b = 1. y = 3x + 1
   2. m=−2, (3,1): 1 = −2(3)+b ⇒ b = 7. y = −2x + 7
   3. m=5, (0,−4): b = −4 directly. y = 5x − 4
   4. m=1/2, (2,6): 6 = (1/2)(2)+b ⇒ b = 5. y = ½x + 5
   5. m=−3/4, (4,0): 0 = (−3/4)(4)+b ⇒ b = 3. y = −¾x + 3
   6. m=4, (−1,−5): −5 = 4(−1)+b ⇒ b = −1. y = 4x − 1

3. x-intercept and y-intercept

   1. y = 2x − 6: y-int: b = −6. x-int: 0 = 2x−6 ⇒ x = 3. y-int (0,−6); x-int (3,0)
   2. y = −3x + 9: y-int: b = 9. x-int: 0 = −3x+9 ⇒ x = 3. y-int (0,9); x-int (3,0)
   3. y = 4x + 8: y-int: b = 8. x-int: 0 = 4x+8 ⇒ x = −2. y-int (0,8); x-int (−2,0)
   4. y = ½x − 2: y-int: b = −2. x-int: 0 = x/2−2 ⇒ x = 4. y-int (0,−2); x-int (4,0)
   5. y = −x + 5: y-int: b = 5. x-int: 0 = −x+5 ⇒ x = 5. y-int (0,5); x-int (5,0)
   6. y = 3x − 12: y-int: b = −12. x-int: 0 = 3x−12 ⇒ x = 4. y-int (0,−12); x-int (4,0)

4. Parallel and perpendicular lines

   1. Parallel to y = 4x−3: m = 4
   2. Perpendicular to y = 4x−3: m = −1/4
   3. Parallel to y = 3x+1 through (2,8): m = 3; 8 = 3(2)+b ⇒ b = 2. y = 3x + 2
   4. Perpendicular to y = 3x+1 through (3,4): m = −1/3; 4 = (−1/3)(3)+b ⇒ b = 5. y = −⅓x + 5
   5. Perpendicular to y = −2x+5 through (−2,1): m = 1/2; 1 = (1/2)(−2)+b ⇒ b = 2. y = ½x + 2
   6. y = 2/3x+4 and y = −3/2x−1: (2/3) × (−3/2) = −1. Perpendicular.

5. Parallel/perpendicular from four points

   1. Line 1: (0,1)(2,5) m=2, y=2x+1. Line 2: (1,3)(3,7) m=2, y=2x+1: Both have m=2 and same equation. They are the same line (coincident).
   2. Line 1: (0,0)(3,6) m=2, y=2x. Line 2: (0,4)(6,1) m=−1/2, y=−x/2+4: 2 × (−1/2) = −1. Perpendicular.
   3. Line 1: (1,2)(4,8) m=2, y=2x. Line 2: (0,5)(6,3) m=−1/3, y=−x/3+5: m₁×m₂ = 2×(−1/3) = −2/3 ≠ −1; m values differ. Neither parallel nor perpendicular.

6. Problem solving

   1. Line through A(1,−2) and B(5,6):
      1. m = (6−(−2))/(5−1) = 2; −2 = 2(1)+b ⇒ b = −4. y = 2x − 4
      2. 0 = 2x−4 ⇒ x = 2. x-intercept: (2, 0)
      3. Perpendicular m = −1/2; through (5,6): 6 = (−1/2)(5)+b ⇒ b = 17/2. y = −½x + 8.5
   2. Adjacent side perpendicular to y=2x−1 through (3,5): Perp. m = −1/2; 5 = (−1/2)(3)+b ⇒ b = 13/2. y = −½x + 6.5
   3. P(k,7) on line through (1,1) and (4,7): m = (7−1)/(4−1) = 2; equation: 1 = 2(1)+b ⇒ b = −1; y = 2x−1. 7 = 2k−1 ⇒ k = 4. k = 4
   4. Route A through (0,10) and (5,0); Route B through (2,3) parallel to A: Route A: m = (0−10)/(5−0) = −2; y = −2x+10. Route B: m = −2; 3 = −2(2)+b ⇒ b = 7; y = −2x+7. Parallel lines never meet (different y-intercepts, same gradient).

7. Lines through the origin and special lines

   1. Line through origin and (4, 10): m = 10/4 = 5/2; b = 0. y = &frac52;x
   2. Through origin, perpendicular to y = 3x: Perp. gradient = −1/3; b = 0. y = −⅓x
   3. Horizontal through (5, −2): y = −2
   4. Vertical through (−3, 7): x = −3
   5. Collinearity of (0,0), (2,5), (6,15): Line through (0,0) and (2,5): m=5/2, so y=5x/2. Check (6,15): y=5(6)/2=15 ✓. Yes, all three points are collinear.

8. Geometric conditions

   1. Triangle A(0,0), B(6,0), C(3,6):
      1. m = (0−0)/(6−0) = 0; b = 0. y = 0 (the x-axis)
      2. m = (6−0)/(3−0) = 2; b = 0. y = 2x
      3. Midpoint of AC = ((0+3)/2,(0+6)/2) = (1.5, 3). Line from B(6,0) to (1.5,3): m = (3−0)/(1.5−6) = 3/(−4.5) = −2/3. 0 = (−2/3)(6)+b ⇒ b=4. y = −⅔x + 4
   2. L₁ through (1,4) and (3,8); L₂ perp. through (3,8):
      1. m = (8−4)/(3−1) = 2; 4=2(1)+b ⇒ b=2. y = 2x + 2
      2. Perp. m = −1/2; 8 = (−1/2)(3)+b ⇒ b = 19/2. y = −½x + 9.5
      3. Set x=0: y = 9.5. y-intercept: (0, 9.5)

9. Intersection points

   1. y=2x+1 and y=−x+7: 2x+1=−x+7 ⇒ 3x=6 ⇒ x=2, y=5. (2, 5)
   2. y=3x−5 and y=x+3: 3x−5=x+3 ⇒ 2x=8 ⇒ x=4, y=7. (4, 7)
   3. y=½x+4 and y=−2x−1: ½x+4=−2x−1 ⇒ &frac52;x=−5 ⇒ x=−2, y=3. (−2, 3). Check: ½(−2)+4=3 ✓; −2(−2)−1=3 ✓
   4. Line A: (0,6)&(3,0) m=−2, y=−2x+6. Line B: (0,0)&(2,4) m=2, y=2x: −2x+6=2x ⇒ 4x=6 ⇒ x=1.5, y=3. Intersection: (1.5, 3)

10. Extended modelling

    1. Road P(2,3) to Q(10,11); pipe perp. through R(8,4):
       1. m=(11−3)/(10−2)=1; 3=1(2)+b ⇒ b=1. y = x + 1
       2. Perp. m=−1; 4=−1(8)+b ⇒ b=12. y = −x + 12
       3. x+1=−x+12 ⇒ 2x=11 ⇒ x=5.5, y=6.5. Intersection: (5.5, 6.5)
    2. A(0,2), B(4,10), C(6,k):
       1. m=(10−2)/(4−0)=2; b=2. y = 2x + 2
       2. Sub (6,k) into y=2x+2: k=2(6)+2=14. k = 14
       3. Gradient AB = 2. For BC perp. to AB, gradient BC = −1/2. m_BC=(k−10)/(6−4)=(k−10)/2=−1/2 ⇒ k−10=−1 ⇒ k = 9