CBSE Class 10 · Chapter 3 · Algebra · 2026-27

Pair of Linear Equations in Two Variables Class 10: MCQ and case study questions

15 multiple-choice questions and 2 case-based questions on the current (rationalised) syllabus, in the style of the board paper's Sections A and E. Try each one first; the answer, a one-line reason and a worked solution open on a tap.

15 MCQs (1 mark each)
2 case studies (4 marks each)
No calculator needed
Free, no sign-in

Multiple-choice questions

Tap an option to check it; the answer and a worked solution open. Section A of the board paper has 18 MCQs of 1 mark each, spread over all chapters.

Q1

·1 mark·Multiple choiceConsistency and number of solutions

How many solutions does the pair $x + 2 y = 5$ , $3 x + 6 y = 12$ have?

Show answer

Answer: (b) None

Why: a₁/a₂ = b₁/b₂ = 1/3 but c₁/c₂ = 5/12, so the lines are parallel.

$\frac{a_{1}}{a_{2}} = \frac{1}{3}$ , $\frac{b_{1}}{b_{2}} = \frac{2}{6} = \frac{1}{3}$ , $\frac{c_{1}}{c_{2}} = \frac{5}{12} \neq \frac{1}{3}$ . The lines are parallel, so there is no solution.

Q2

·1 mark·Multiple choiceAlgebraic solution (substitution / elimination)

The solution of $2 x + y = 11$ and $x - y = 1$ is

Show answer

Answer: (c)

x = 4, y = 3

Why: Adding the equations gives 3x = 12.

Adding: $3 x = 12 \Rightarrow x = 4$ . Then $y = x - 1 = 3$ . Check: $2 (4) + 3 = 11$ .

Q3

·1 mark·Multiple choiceConsistency and number of solutions

For which value of $k$ do $3 x - y = 2$ and $6 x + k y = 4$ have infinitely many solutions?

Show answer

Answer: (b)

- 2

Why: Need 3/6 = −1/k = 2/4, so −1/k = 1/2.

$\frac{3}{6} = \frac{- 1}{k} = \frac{2}{4}$ . From $\frac{- 1}{k} = \frac{1}{2}$ , $k = - 2$ .

Q4

·1 mark·Multiple choiceConsistency and number of solutions

The lines $4 x + m y = 7$ and $2 x + 3 y = 5$ intersect at exactly one point when

Show answer

Answer: (d)

m \neq 6

Why: A unique solution needs a₁/a₂ ≠ b₁/b₂, i.e. 2 ≠ m/3.

$\frac{a_{1}}{a_{2}} = \frac{4}{2} = 2$ . Unique solution when $\frac{m}{3} \neq 2$ , i.e. $m \neq 6$ . (At $m = 6$ the lines are parallel: $\frac{c_{1}}{c_{2}} = \frac{7}{5} \neq 2$ .)

Q5

·1 mark·Multiple choiceGraphical representation

The graph of $2 x - 5 y = 10$ meets the $x$ -axis at

Show answer

Answer: (c)

(5, 0)

Why: On the x-axis y = 0, so 2x = 10.

Put $y = 0$ : $2 x = 10 \Rightarrow x = 5$ . The point is $(5, 0)$ .

Q6

·1 mark·Multiple choiceAlgebraic solution (substitution / elimination)

If $x + y = 7$ and $2 x - 3 y = - 1$ , then $x - y$ equals

Show answer

Answer: (a)

1

Why: Substitute x = 7 − y: 14 − 5y = −1 gives y = 3, x = 4.

$x = 7 - y \Rightarrow 2 (7 - y) - 3 y = - 1 \Rightarrow 14 - 5 y = - 1 \Rightarrow y = 3$ , $x = 4$ . So $x - y = 1$ .

Q7

·1 mark·Multiple choiceConsistency and number of solutions

The lines represented by $3 x - 4 y = 8$ and $9 x - 12 y = 24$ are

Show answer

Answer: (c) coincident

Why: All three ratios are equal: 3/9 = −4/−12 = 8/24 = 1/3.

$\frac{3}{9} = \frac{- 4}{- 12} = \frac{8}{24} = \frac{1}{3}$ . The second equation is $3$ times the first, so the lines coincide.

Q8

·1 mark·Multiple choiceConsistency and number of solutions

Which statement is true for the pair $5 x + 2 y = 9$ and $2 x - y = 0$ ?

Show answer

Answer: (b) The lines cross once, so the pair has exactly one solution

Why: 5/2 ≠ 2/(−1), so the lines intersect: one solution (1, 2).

$\frac{a_{1}}{a_{2}} = \frac{5}{2}$ , $\frac{b_{1}}{b_{2}} = - 2$ ; these differ, so there is exactly one solution: $y = 2 x$ , $9 x = 9$ , $x = 1$ , $y = 2$ .

Q9

·1 mark·Multiple choiceSituational problems

The sum of two numbers is $50$ and their difference is $14$ . The larger number is

Show answer

Answer: (a)

32

Why: Add x + y = 50 and x − y = 14: 2x = 64.

$x + y = 50$ , $x - y = 14$ . Adding, $2 x = 64$ , so $x = 32$ (and $y = 18$ ).

Q10

·1 mark·Multiple choiceSituational problems

$3$ pens and $2$ notebooks cost ₹ $110$ , while $2$ pens and $3$ notebooks cost ₹ $115$ . The cost of one notebook is

Show answer

Answer: (d) ₹

25

Why: Add: 5(p + n) = 225; subtract: n − p = 5.

$3 p + 2 n = 110$ , $2 p + 3 n = 115$ . Adding: $p + n = 45$ ; subtracting: $n - p = 5$ . So $n = 25$ , $p = 20$ .

Q11

·1 mark·Multiple choiceConsistency and number of solutions

For which value of $a$ do $a x + 4 y = 8$ and $5 x + 10 y = 20$ have infinitely many solutions?

Show answer

Answer: (a)

2

Why: b₁/b₂ = c₁/c₂ = 2/5, so a/5 must also be 2/5.

$\frac{4}{10} = \frac{8}{20} = \frac{2}{5}$ , so $\frac{a}{5} = \frac{2}{5} \Rightarrow a = 2$ .

Q12

·1 mark·Multiple choiceGraphical representation

The lines $y = 2 x$ and $y = - x + 6$ meet at the point

Show answer

Answer: (b)

(2, 4)

Why: 2x = −x + 6 gives x = 2, then y = 4.

$2 x = - x + 6 \Rightarrow 3 x = 6 \Rightarrow x = 2$ , $y = 4$ . The point is $(2, 4)$ .

Q13

·1 mark·Multiple choiceGraphical representation

The lines $x - y = 1$ and $x + y = 5$ and the $y$ -axis enclose a triangle. Its area (in square units) is

Show answer

Answer: (c)

9

Why: Vertices (0, −1), (0, 5) and (3, 2): base 6 on the y-axis, height 3.

The lines meet at $(3, 2)$ and cut the $y$ -axis at $(0, - 1)$ and $(0, 5)$ . Area $= \frac{1}{2} \times 6 \times 3 = 9$ .

Q14

·1 mark·Multiple choiceSituational problems

Riya is $4$ years older than twice her brother's age, and the sum of their ages is $25$ years. Riya's age is

Show answer

Answer: (a)

18

years

Why: r = 2b + 4 and r + b = 25 give 3b + 4 = 25.

$r = 2 b + 4$ , $r + b = 25 \Rightarrow 3 b + 4 = 25 \Rightarrow b = 7$ , so $r = 18$ .

Q15

·1 mark·Multiple choiceConsistency and number of solutions

The pair $2 x - 3 y = 4$ and $6 x - 9 y = c$ is inconsistent when

Show answer

Answer: (b)

c \neq 12

Why: a₁/a₂ = b₁/b₂ = 1/3, so no solution exactly when 4/c ≠ 1/3.

$\frac{2}{6} = \frac{- 3}{- 9} = \frac{1}{3}$ . The pair has no solution when $\frac{4}{c} \neq \frac{1}{3}$ , i.e. $c \neq 12$ ; at $c = 12$ the lines coincide.

Case study questions

Section E of the board paper has three case-based questions of 4 marks: a real-life passage, then parts of 1, 1 and 2 marks, with a choice (OR) on the 2-mark part. All Class 10 case study questions by chapter.

Case study 1: School canteen (4 marks)

On Monday a group of friends bought $4$ samosas and $3$ glasses of juice from the school canteen for ₹ $130$ . On Tuesday another group bought $2$ samosas and $5$ glasses of juice for ₹ $170$ . Let one samosa cost ₹ $x$ and one glass of juice cost ₹ $y$ .

(i) Write the pair of linear equations that describes the two purchases. [1 mark]

Show answer

Answer:

4 x + 3 y = 130

2 x + 5 y = 170

4 x + 3 y = 130

and

2 x + 5 y = 170

. A1

(ii) Are the lines representing these equations intersecting, parallel or coincident? Give a reason. [1 mark]

Show answer

Answer: Intersecting

\frac{a_{1}}{a_{2}} = 2 \neq \frac{b_{1}}{b_{2}} = \frac{3}{5}

, so the lines intersect (unique solution). A1

(iii) Find the cost of one samosa and of one glass of juice. [2 marks]

Show answer

Answer: Samosa ₹

10

, juice ₹

30

Double the second equation:

4 x + 10 y = 340

; subtract the first:

7 y = 210

M1,

y = 30

x = 10

. A1

OR How much would

6

samosas and

2

glasses of juice cost? [2 marks]

Show answer

Answer: ₹

120

Solving gives

x = 10

y = 30

M1;

6 \times 10 + 2 \times 30 = ₹ 120

. A1

Case study 2: Fencing a park (4 marks)

A residents' welfare association is fencing a rectangular park. The fence around it is $84$ m long, and the length of the park is $6$ m more than twice its width. Let the length be $l$ m and the width be $w$ m.

(i) Write a pair of linear equations in

l

and

w

. [1 mark]

Show answer

Answer:

l + w = 42

l - 2 w = 6

2 (l + w) = 84

, i.e.

l + w = 42

, and

l = 2 w + 6

, i.e.

l - 2 w = 6

. A1

(ii) Find the width of the park. [1 mark]

Show answer

Answer:

12

Substitute

l = 2 w + 6

3 w + 6 = 42 \Rightarrow w = 12

m. A1

(iii) Find the length and the area of the park. [2 marks]

Show answer

Answer:

30

360 m^{2}

l = 2 (12) + 6 = 30

m A1; area

= 30 \times 12 = 360 m^{2}

. A1

OR The association later adds

4

m to the length and removes

2

m from the width. Find the new area. [2 marks]

Show answer

Answer:

340 m^{2}

New length

34

m, new width

10

m M1; area

= 340 m^{2}

. A1

Next steps for Pair of Linear Equations in Two Variables

This free set is separate from the chapter's question bank. On the Pair of Linear Equations in Two Variables chapter page the revision notes and the first step of the Route to 95 are free for everyone. CBSE Pro adds all 51 questions in the chapter bank (short and long answers, assertion–reason and more case studies) with their full step-marking schemes, the Route to 95 checkpoints with your progress saved, Skill Builders, the full common-mistakes library and AI marking of your written working.

Pair of Linear Equations in Two Variables: important questions What Pro includes

Original questions written by CBSE Math Revision for the CBSE 2026-27 syllabus; not taken from NCERT, NCERT Exemplar or CBSE papers. Every answer was checked twice, the second time by an independent reviewer. CBSE Math Revision is independent and not affiliated with CBSE or NCERT. Spotted a slip? Tell us and it goes in the corrections log.