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CBSE · Class 10 · Mathematics

Pair of Linear Equations in Two Variables

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
The solution of 2x + 2y = −8 and 4x − 5y = −16 is:
  1. (−4, 0)
  2. (−3, 0)
  3. (4, 0)
  4. (0, −4)
2.
The pair 5x + 5y = 12 and 20x + 20y = 48 has:
  1. a unique solution
  2. no solution
  3. infinitely many solutions
  4. exactly two solutions
3.
For what value of k does 5x − 5y = −9 and 15x + ky = −26 have no solution?
  1. k = 15
  2. k = −5
  3. k = −15
  4. k = −14
4.
If x + y = 21 and x − y = 11, then (x, y) is:
  1. (16, 5)
  2. (21, 11)
  3. (16, −5)
  4. (5, 16)
5.
Two lines that intersect at one point give a pair of equations with:
  1. a unique solution
  2. exactly two solutions
  3. no solution
  4. infinitely many solutions
6.
If 4x − 5y = 9 and 5x − y = 27, then x + y equals:
  1. 10
  2. 3
  3. 18
  4. 9
7.
For what value of k does 3x + y = −8 and 12x + 4y = k have infinitely many solutions?
  1. k = 32
  2. k = −8
  3. k = −4
  4. k = −32
8.
Assertion (A): The pair 3x + y = 1 and 6x + 2y = 3 has a unique solution.
Reason (R): A pair of linear equations has a unique solution when a1/a2 ≠ b1/b2.
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Section B — Short answer8 × 2 = 16 marks
9.
Solve by substitution: x + y = 4 and 6x + 3y = 36
10.
Solve by elimination: x + y = 15 and 5x − 7y = 3
11.
Solve: x/3 + y/3 = 14 and x/2 − y/2 = −3
12.
Find k so that 2x + 2y = −3 and 4x + ky = −3 have no solution.
13.
Solve: 12/x + 24/y = 7 and 16/x + 12/y = 6
14.
Check whether 5x + y = 39 and x + 2y = 6 is consistent. If it is, solve it.
15.
Solve: 93x + 92y = 929 and 92x + 93y = 921
16.
The sum of two numbers is 68 and their difference is 26. Find the numbers.
Section C — Word problems4 × 4 = 16 marks
17.
4 pencils and 5 erasers cost Rs 55. 3 pencils and 4 erasers cost Rs 42. Find the cost of one pencil and one eraser.
18.
Five years ago, Meera was 2 times as old as Zoya. The sum of their present ages is 70 years. Find their present ages.
19.
A boat goes 100 km downstream in 5 hours and 32 km upstream in 4 hours. Find the speed of the boat in still water and of the stream.
20.
A taxi charges a fixed amount plus a charge per km. For 11 km the fare is Rs 161, and for 16 km it is Rs 216. Find the fixed charge and the charge per km.
AmbifulWorksheets
Practice that never repeats
CBSE · Class 10 · Mathematics

Answer key

Pair of Linear Equations in Two Variables · AW-10M-PLE-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(a) (−4, 0)solve: x = −4, y = 0
2(c) infinitely many solutionsa1/a2 = b1/b2 = c1/c2
3(c) k = −15a1/a2 = 1/3, so b2 = 3 × (−5) = −15
4(a) (16, 5)adding: 2x = 32, x = 16; y = 5
5(a) a unique solution—
6(d) 9x = 6, y = 3
7(d) k = −32the second equation must be 4 times the first
8(d) A is false, but R is true.A is false: a1/a2 = b1/b2 = 1/2: the lines are parallel. R is true.
9x = 8, y = −4x = 4 − y; substitute into the second
10x = 9, y = 6make the coefficients of one variable equal, then subtract
11x = 18, y = 24multiply the first equation by 3 and the second by 2 to clear the fractions, then eliminate
12k = 4a1/a2 = 1/2, so k = 2 × 2 = 4 (and c1/c2 ≠ 1/2)
13x = 4, y = 6put u = 1/x, v = 1/y: 12u + 24v = 7; u = 1/4, v = 1/6
14Consistent; x = 8, y = −1a1/a2 ≠ b1/b2, so there is a unique solution
15x = 9, y = 1add: x + y = 10; subtract: x − y = 8
1647 and 21x + y = 68, x − y = 26; 2x = 94
17pencil Rs 10, eraser Rs 34x + 5y = 55; 3x + 4y = 42
18Meera 45 years, Zoya 25 yearsx − 5 = 2(y − 5); x + y = 70
19boat 14 km/h, stream 6 km/hu + v = 20, u − v = 8
20fixed Rs 40, Rs 11 per kmx + 11y = 161; x + 16y = 216; subtract: 5y = 55