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Gyaan Maths
CBSE · Class 6-10
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CBSE Maths · Class 10

Chapter 3: Pair of Linear Equations in Two Variables

Name: ________________________
Date: ______________
Marks: ____ / 10

Graphical and algebraic methods (substitution, elimination, cross-multiplication) to solve two linear equations in two variables.

Questions

  1. Solve 2x+3y=11,  2x−4y=−242x + 3y = 11,\; 2x - 4y = -24.
  2. Find kk for which 2x+3y=42x + 3y = 4 and kx+6y=8kx + 6y = 8 have infinitely many solutions.
  3. Solve 3x−5y=−1,  x−y=−13x - 5y = -1,\; x - y = -1 by substitution.
  4. Solve x+y=7,  x−y=3x + y = 7,\; x - y = 3.
  5. Solve 2x+y=5,  3x+2y=82x + y = 5,\; 3x + 2y = 8 by cross-multiplication.
  6. For what kk does 3x+y=1,(2k−1)x+(k−1)y=2k+13x + y = 1, (2k-1)x + (k-1)y = 2k+1 have no solution?
  7. Pen cost pp, book bb: 5p+6b=250,  6p+5b=2405p + 6b = 250,\; 6p + 5b = 240. Find p,bp, b.
  8. 2x+3y=13,  5x−4y=−2\dfrac{2}{x} + \dfrac{3}{y} = 13,\; \dfrac{5}{x} - \dfrac{4}{y} = -2.
  9. Father is 3 times son's age; in 12 years he'll be twice. Present ages?
  10. Half the perimeter of a rectangle is 3636 m and length exceeds width by 44 m. Find dimensions.

Extra Practice (Optional)

  • Solve 3x+2y=8,x−y=13x + 2y = 8, x - y = 1.
  • kk for no solution of 2x−3y=7,  4x−6y=k2x - 3y = 7,\; 4x - 6y = k.
  • Digit-swap problem: 2-digit number, digits sum 99, reversed exceeds original by 2727.
  • Boat downstream/upstream problem.
  • 1x+y+2x−y=2\dfrac{1}{x+y} + \dfrac{2}{x-y} = 2, 2x+y−1x−y=3\dfrac{2}{x+y} - \dfrac{1}{x-y} = 3.

Answer Key (For Teachers)

  1. Answer: (−2,5)(-2, 5).
    Working: Subtract: 7y=35⇒y=57y = 35 \Rightarrow y = 5. → 2x+15=11⇒x=−22x + 15 = 11 \Rightarrow x = -2.
  2. Answer: k=4k = 4.
    Working: 2k=36=48\dfrac{2}{k} = \dfrac{3}{6} = \dfrac{4}{8}.
  3. Answer: (−2,−1)(-2, -1).
    Working: From 2nd: x=y−1x = y - 1. → 3(y−1)−5y=−1⇒y=−1,  x=−23(y-1) - 5y = -1 \Rightarrow y = -1,\; x = -2.
  4. Answer: (5,2)(5, 2).
    Working: Add: 2x=102x = 10; sub back for yy.
  5. Answer: (2,1)(2, 1).
    Working: Set ci=−constantc_i = -\text{constant}; apply formula. → Compute x/2=y/1=1/1x/2 = y/1 = 1/1.
  6. Answer: k=2k = 2.
    Working: No sol: 32k−1=1k−1≠12k+1\dfrac{3}{2k-1} = \dfrac{1}{k-1} \ne \dfrac{1}{2k+1}. → From first: 3(k−1)=2k−1⇒k=23(k-1) = 2k-1 \Rightarrow k = 2; check ratio for constant differs.
  7. Answer: p=20,  b=25p = 20,\; b = 25.
    Working: Solve pair: gives p=20,b=25p = 20, b = 25.
  8. Answer: x=1/2,y=1/3x = 1/2, y = 1/3.
    Working: Sub u=1/x,v=1/yu = 1/x, v = 1/y: solve 2u+3v=13,5u−4v=−22u+3v=13, 5u-4v=-2; u=2,v=3u = 2, v = 3.
  9. Answer: 12,  3612,\; 36.
    Working: Set s,3ss, 3s; 3s+12=2(s+12)3s+12 = 2(s+12).
  10. Answer: l=20l = 20 m, b=16b = 16 m.
    Working: l+b=36,  l−b=4l + b = 36,\; l - b = 4. → Add: 2l=40⇒l=202l = 40 \Rightarrow l = 20, b=16b = 16.
Gyaan Maths · CBSE Class 10 · Chapter 3
Gyaan Maths — CBSE Class 6-10, built for board-toppers.
v1.1 · Feb 2026