Elimination Method — Solutions

Click any answer to watch the solution video.

1. Direct elimination (add or subtract)

   1. 3x+y=11 and x+y=5: Subtract: 2x=6 → x=3, y=2
   2. 4x+3y=19 and 2x+3y=13: Subtract: 2x=6 → x=3, y=7/3 … Check: 4(3)+3(7/3)=12+7=19 ✓
   3. 5x+2y=16 and 3x−2y=0: Add: 8x=16 → x=2, y=3
   4. x+4y=14 and x−4y=−2: Add: 2x=12 → x=6, y=2
   5. 6x+y=20 and 6x−y=10: Add: 12x=30 → x=2.5, y=5
   6. 2x+5y=24 and 2x+y=8: Subtract: 4y=16 → y=4, x=2
   7. 7x−3y=25 and x+3y=7: Add: 8x=32 → x=4, y=1
   8. 4x+7y=30 and 4x+3y=18: Subtract: 4y=12 → y=3, x=9/4

2. Multiply then eliminate

   1. 2x+y=9, 3x+2y=14: Multiply eq1 ×2: 4x+2y=18. Subtract eq2: x=4, y=1
   2. 3x+2y=16, x+y=7: Multiply eq2 ×2: 2x+2y=14. Subtract from eq1: x=2, y=5
   3. 4x+3y=23, 2x+y=9: Multiply eq2 ×3: 6x+3y=27. Subtract from eq1: −2x=−4 → x=2. Sub: y=5. Check: 4(2)+3(5)=23 ✓
   4. 3x+4y=26, 2x+y=10: Multiply eq2 ×4: 8x+4y=40. Subtract eq1: 5x=14 → x=14/5… Alt: multiply eq2 ×3 and eq1 ×2: 6x+8y=52, 6x+3y=30. Subtract: 5y=22 → y=22/5, x=14/5
   5. 5x+2y=29, 3x+y=16: Multiply eq2 ×2: 6x+2y=32. Subtract eq1: x=3, y=7
   6. 2x+3y=13, 3x+2y=12: Multiply eq1 ×3, eq2 ×2: 6x+9y=39, 6x+4y=24. Subtract: 5y=15 → y=3, x=2

3. Mixed method and special cases

   1. y=3x−1 and 2x+y=9: Best: substitution (y isolated). 2x+(3x−1)=9 → 5x=10 → x=2, y=5
   2. 4x−3y=1 and 5x−3y=−2: Best: elimination (3y coefficients match). Subtract eq1 from eq2: x=−3. Sub: 4(−3)−3y=1 → y=−13/3
   3. 3x+2y=12 and 6x+4y=24: Multiply eq1 ×2: 6x+4y=24. Subtract eq2: 0=0. Infinite solutions — both equations describe the same line.
   4. 2x+y=5 and 4x+2y=15: Multiply eq1 ×2: 4x+2y=10. Subtract eq2: 0=−5. No solution — the lines are parallel and never intersect.

4. Problem solving with elimination

   1. Sum 23, difference 7: x+y=23, x−y=7. Add: 2x=30 → x=15, y=8
   2. Mobile plans equal cost: 20+0.3m=10+0.5m → 10=0.2m → m=50 minutes. Cost = 20+0.3(50)=$35
   3. Cinema tickets: a+s=80, 15a+10s=1050. Multiply eq1 ×10: 10a+10s=800. Subtract: 5a=250 → a=50 adults, s=30 students
   4. Sprinter wind problem: 400/(s+w)=44 → s+w=400/44≈9.09; 400/(s−w)=56 → s−w≈7.14. Add: 2s≈16.23 → s≈8.12 m/s, w≈0.97 m/s

5. Multiply both equations, then eliminate

   1. 3x+5y=23, 4x+3y=20: Multiply eq1 ×4: 12x+20y=92. Multiply eq2 ×3: 12x+9y=60. Subtract: 11y=32 → y=32/11. Back-sub into eq2: 4x+3(32/11)=20 → 4x=20−96/11=124/11 → x=31/11. Check: 3(31/11)+5(32/11)=93/11+160/11=253/11=23 ✓
   2. 5x+4y=33, 3x+2y=19: Multiply eq2 ×2: 6x+4y=38. Subtract eq1: x=5. Sub: 3(5)+2y=19 → 2y=4 → y=2. Check: 5(5)+4(2)=25+8=33 ✓
   3. 2x+7y=39, 5x+3y=27: Multiply eq1 ×5: 10x+35y=195. Multiply eq2 ×2: 10x+6y=54. Subtract: 29y=141 → y=141/29≈4.86. Sub: 2x+7(141/29)=39 → 2x=39−987/29=(1131−987)/29=144/29 → x=72/29≈2.48

6. Clearing fractions and decimals before eliminating

   1. ½x+⅓y=4, ¼x+½y=3: Multiply eq1 ×6: 3x+2y=24. Multiply eq2 ×4: x+2y=12. Subtract: 2x=12 → x=6. Sub: 6+2y=12 → y=3. Check: ½(6)+⅓(3)=3+1=4 ✓
   2. 0.3x+0.2y=2.6, 0.5x−0.4y=0.2: Multiply both by 10: 3x+2y=26, 5x−4y=2. Multiply eq1 ×2: 6x+4y=52. Add eq2: 11x=54 → x=54/11≈4.91. Sub: 3(54/11)+2y=26 → 2y=26−162/11=124/11 → y=62/11≈5.64

7. Number and digit problems

   1. Two-digit number, digit sum 9, reversed is 27 more: t+u=9. Reversed: (10u+t)−(10t+u)=27 → 9u−9t=27 → u−t=3. Add: 2u=12 → u=6, t=3. Original number = 36.
   2. Difference 14, 3(smaller)+2(larger)=93: Let a>b. a−b=14. 3b+2a=93. From eq1: a=b+14. Sub: 3b+2(b+14)=93 → 5b=65 → b=13, a=27. Check: 3(13)+2(27)=39+54=93 ✓
   3. 2x+3y=38, y−x=4: From eq2: y=x+4. Sub: 2x+3(x+4)=38 → 5x=26 → x=26/5=5.2, y=9.2. Check: 2(5.2)+3(9.2)=10.4+27.6=38 ✓

8. Finance and investment problems

   1. Investment $8000, 4% and 6%, total interest $400: Let a=amount in Account A, b=amount in Account B. a+b=8000; 0.04a+0.06b=400. Multiply eq1 ×0.04: 0.04a+0.04b=320. Subtract: 0.02b=80 → b=$4000, a=$4000.
   2. 120 items sold, sausage rolls $2.50, meat pies $3.80, total $366: Let s=sausage rolls, p=meat pies. s+p=120; 2.5s+3.8p=366. Multiply eq1 ×2.5: 2.5s+2.5p=300. Subtract: 1.3p=66 → p=66/1.3≈50.77. Recheck: using exact: 1.3p=66 → p=660/13≈50.8. Since answers must be whole numbers, re-examine: 2.5s+3.8(120−s)=366 → 2.5s+456−3.8s=366 → −1.3s=−90 → s=90/1.3≈69.2. Accept s=69, p=51 (verify: 2.5(69)+3.8(51)=172.50+193.80=$366.30 ≈ close; exact values are non-integer, accept as set).

9. Rectangle perimeter investigation

   1. Original: 2l+2w=64 → l+w=32
   2. Modified (length +5, width −2): 2(l+5)+2(w−2)=64 → 2l+10+2w−4=64 → 2l+2w=58 → l+w=29. (Different from 32.)
   3. Solve: We have l+w=32 and l+w=29. Subtracting: 0=3. This is a false statement — no solution. Geometrically: the two conditions are inconsistent. A rectangle cannot have the same perimeter after those changes; the problem setup contains a contradiction.

10. Two-pipe fill problem

    1. Equations: ¹⁄_a+¹⁄_b=¹⁄₄ and b−a=6 (Pipe A is 6 hours faster, so b=a+6).
    2. Solve: Let A=1/a, B=1/b: A+B=1/4. From b=a+6, substitute directly: 1/a+1/(a+6)=1/4. Multiply through by 4a(a+6): 4(a+6)+4a=a(a+6) → 8a+24=a²+6a → a²−2a−24=0 → (a−6)(a+4)=0 → a=6 hours.
    3. Answer: Pipe A fills the tank in 6 hours alone. Pipe B fills it in b=6+6=12 hours alone. Verify: 1/6+1/12=2/12+1/12=3/12=1/4 ✓