Class 10 Maths Chapter 3: Pair of Linear Equations in Two Variables worksheet with answer key
A 20-mark worksheet on Pair of Linear Equations in Two Variables for about 40 minutes of class or homework: 5 multiple-choice, 3 very short answer and 3 short answer questions, written from the NCERT chapter and checked by a second reviewer. Print the PDF for the class; the answer key carries the marking scheme. Teachers may photocopy it freely.
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Answer: (d) 3
3x − 4(7 − 2x) = 5 gives 3x − 28 + 8x = 5, so 11x = 33 and x = 3 (then y = 1).
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Answer: (b) exactly one solution
a₁/a₂ = 3/4 and b₁/b₂ = 5/(−2) = −5/2. Since 3/4 ≠ −5/2, the lines intersect, so there is exactly one solution.
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Answer: (c) 2
Two lines either meet in one point, never meet, or coincide. So a pair has one solution, none, or infinitely many; exactly two is impossible.
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Answer: (b) 6x − 8y + 12 = 0
6x − 8y + 12 = 0 is 2(3x − 4y + 6) = 0: a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2. Option (a) and option (d) have the x and y ratios equal but a different constant ratio (parallel); (c) has 3/4 ≠ −4/−3 (intersecting).
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Answer: (a) 3
The y-coefficients are 3 and −3, opposite in sign, so adding removes y: 9x = 27, x = 3.
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Unique solution needs a₁/a₂ ≠ b₁/b₂: 1/5 ≠ 2/k, i.e. k ≠ 10. So every real k except 10.
- uses the condition a₁/a₂ ≠ b₁/b₂ (1)
- k ≠ 10 (1)
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Multiply both by 10: 3x + 5y = 19 and 7x − 2y = 17. Multiply by 2 and 5: 6x + 10y = 38, 35x − 10y = 85. Add: 41x = 123, x = 3. Then 5y = 19 − 9, y = 2.
- clears decimals to 3x + 5y = 19 and 7x − 2y = 17 (1)
- x = 3, y = 2 (1)
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From the first, x = 10 − 3y. Into the second: 2(10 − 3y) − y = 6, so 20 − 7y = 6, y = 2. Then x = 10 − 6 = 4. Solution x = 4, y = 2.
- writes x = 10 − 3y and substitutes into the other equation, getting y = 2 (1)
- x = 4 (1)
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Let the fixed charge be ₹x and the rate ₹y. Only hours after the first are charged: x + 3y = 110, x + 6y = 170. Subtract: 3y = 60, y = 20; x = 50. Fixed charge ₹50, ₹20 per extra hour. Check: 50 + 60 = 110, 50 + 120 = 170.
- counts charged hours: x + 3y = 110 and x + 6y = 170 (1)
- solves y = 20 (1)
- x = 50, and verifies against both bills (1)
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Let the number be 10x + y. 10x + y = 7(x + y) gives 3x = 6y, x = 2y. Case x − y = 3: y = 3, x = 6, number 63. Case y − x = 3: y − 2y = 3, y = −3, not a digit. So the number is 63 (7 × 9 = 63).
- forms 10x + y = 7(x + y), so x = 2y (1)
- takes both cases x − y = 3 and y − x = 3 (1)
- rejects the negative digit; number 63 (1)
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f + d = 50 and f − 5 = 7(d − 5), i.e. f = 7d − 30. Then 8d − 30 = 50, d = 10, f = 40. Check: five years ago 35 and 5, and 35 = 7 × 5.
- both equations, subtracting 5 from both ages (1)
- solves d = 10 (1)
- father 40 years, daughter 10 years (1)