Topic Review — Simultaneous Equations — Solutions

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1. Substitution method

   1. y=x+5, 2x+y=14: 2x+(x+5)=14 → 3x=9 → x=3, y=8
   2. y=4x−3, x+y=12: x+4x−3=12 → 5x=15 → x=3, y=9
   3. x=3y, x+2y=15: 3y+2y=15 → 5y=15 → y=3, x=9
   4. x+y=10, 3x+2y=26: From (1): y=10−x. Sub: 3x+2(10−x)=26 → x=6, y=4

2. Elimination method

   1. 5x+2y=21 and 3x+2y=13: Subtract: 2x=8 → x=4, y=0.5
   2. 4x−y=9 and 2x+y=9: Add: 6x=18 → x=3, y=3
   3. 3x+4y=22 and x+2y=10: Multiply eq2 ×2: 2x+4y=20. Subtract: x=2, y=4
   4. 2x+3y=17 and 3x+2y=18: Multiply eq1 ×3, eq2 ×2: 6x+9y=51, 6x+4y=36. Subtract: 5y=15 → y=3, x=4

3. Choose your method

   1. y=2x+1 and 3x−y=4: Best: substitution (y isolated). 3x−(2x+1)=4 → x=5, y=11
   2. 5x+3y=28 and 5x−y=8: Best: elimination (5x coefficients match). Subtract: 4y=20 → y=5, x=13/5. Check: 5(2.6)+3(5)=13+15=28 ✓
   3. 3x+y=7 and 6x+2y=10: Multiply eq1 ×2: 6x+2y=14. Compare with eq2: 6x+2y=10. Same left side, different right side → 14 ≠ 10 → No solution. Lines are parallel.

4. Interpreting and verifying

   1. Jackson’s claim x=4, y=4: Eq1: 4+4=8 ✓ Eq2: 2(4)−4=4 ✓. Jackson is correct. Solution: x=4, y=4
   2. Point (2,3): Eq1: 3=2(2)−1=3 ✓ Eq2: 2+3=5 ≠ 6 ✗. (2,3) is NOT the solution. Correct: y=2x−1 sub into x+y=6: x+2x−1=6 → x=7/3, y=11/3

5. Real-world problems

   1. Sum 32, difference 10: x+y=32, x−y=10. Add: 2x=42 → x=21, y=11
   2. Tuck shop pies and sandwiches: p+s=55, 4p+3s=190. Multiply eq1 ×3: 3p+3s=165. Subtract: p=25 pies, s=30 sandwiches
   3. Landscaper call-out fee and hourly rate: f+2h=110, f+5h=200. Subtract: 3h=90 → h=$30/hr, f=$50 call-out fee
   4. Sheep and chickens: s+c=25 (animals); 4s+2c=76 (legs). Multiply eq1 ×2: 2s+2c=50. Subtract: 2s=26 → s=13 sheep, c=12 chickens
   5. Triangle angles: p+q+50=180 → p+q=130; p=2q. Sub: 2q+q=130 → 3q=130 → q=43.3°, p=86.7°