Question 1

The pair of equations $x+2y=5$ and $3x+6y=15$ has:

Accepted Answer

**Answer: (c) infinitely many solutions** — $\dfrac{a_1}{a_2}=\dfrac{1}{3},\ \dfrac{b_1}{b_2}=\dfrac{2}{6}=\dfrac{1}{3},\ \dfrac{c_1}{c_2}=\dfrac{5}{15}=\dfrac{1}{3}$. All three ratios are equal, so the lines are coincident.

Question 2

The graph below shows two lines representing a pair of linear equations. The pair is: ```svg ```

Accepted Answer

**Answer: (a) consistent with a unique solution** — The two lines intersect at exactly one point $(2,3)$, so the system is consistent and has a unique solution.

Question 3

The value of $k$ for which the pair $2x+3y=7$ and $(k-1)x+(k+2)y=3k$ has infinitely many solutions is:

Accepted Answer

**Answer: (c) $k=7$** — For infinitely many solutions $\dfrac{2}{k-1}=\dfrac{3}{k+2}=\dfrac{7}{3k}$. From $\dfrac{2}{k-1}=\dfrac{3}{k+2}$: $\ 2(k+2)=3(k-1)\Rightarrow k=7$. This also satisfies $\dfrac{7}{3k}=\dfrac{7}{21}=\dfrac{1}{3}=\dfrac{2}{6}$. ✓

Question 4

The pair of equations $3x-y=5$ and $6x-2y=k$ has **no solution** for:

Accepted Answer

**Answer: (b) all $k
eq10$** — Here $\dfrac{a_1}{a_2}=\dfrac{3}{6}=\dfrac{1}{2}$ and $\dfrac{b_1}{b_2}=\dfrac{-1}{-2}=\dfrac{1}{2}$ are equal. For no solution we need $\dfrac{c_1}{c_2}
eq\dfrac{1}{2}$, i.e. $\dfrac{5}{k}
eq\dfrac{1}{2}\Rightarrow k
eq10$.

Question 5

**Assertion (A):** The pair of equations $x+2y-4=0$ and $2x+4y-12=0$ has no solution. **Reason (R):** A pair of linear equations has no solution when $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}
eq\dfrac{c_1}{c_2}$.

Accepted Answer

**Answer: (a)** Both are true and R explains A. The ratios are $\dfrac{1}{2}=\dfrac{2}{4}=\dfrac{1}{2}$ but $\dfrac{c_1}{c_2}=\dfrac{-4}{-12}=\dfrac{1}{3}
eq\dfrac{1}{2}$, so the lines are parallel and the system has no solution.

Question 6

For what value of $k$ will the lines $3x+2ky=2$ and $2x+5y+1=0$ be parallel?

Accepted Answer

For parallel lines $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}$ (with the constant ratio unequal). Thus $\dfrac{3}{2}=\dfrac{2k}{5}\Rightarrow 4k=15\Rightarrow k=\dfrac{15}{4}$.

Question 7

Solve the pair of equations $2x+y=23$ and $4x-y=19$. Hence find the value of $5y-2x$.

Accepted Answer

Adding the two equations: $6x=42\Rightarrow x=7$. Then $y=23-2(7)=9$. So $5y-2x=5(9)-2(7)=45-14=31$.

Question 8

A fraction becomes $\dfrac{1}{3}$ when $1$ is subtracted from its numerator, and it becomes $\dfrac{1}{4}$ when $8$ is added to its denominator. Find the fraction. *(CBSE 2023)*

Accepted Answer

Let the fraction be $\dfrac{x}{y}$. $\dfrac{x-1}{y}=\dfrac{1}{3}\Rightarrow 3x-y=3\quad	ext{...(i)}$ $\dfrac{x}{y+8}=\dfrac{1}{4}\Rightarrow 4x-y=8\quad	ext{...(ii)}$ Subtracting (i) from (ii): $x=5$. Then $y=3x-3=12$. The fraction is $\dfrac{5}{12}$.

Question 9

Solve for $x$ and $y$: $\ \dfrac{2}{x}+\dfrac{3}{y}=13$ and $\dfrac{5}{x}-\dfrac{4}{y}=-2$.

Accepted Answer

Let $u=\dfrac{1}{x}$ and $v=\dfrac{1}{y}$ : $\ 2u+3v=13\ \text{...(i)}$ , $\ 5u-4v=-2\ \text{...(ii)}$ .

Multiply (i) by $4$ and (ii) by $3$ : $\ 8u+12v=52$ and $15u-12v=-6$ . Adding: $23u=46\Rightarrow u=2$ .

From (i): $3v=13-4=9\Rightarrow v=3$ .

So $x=\dfrac{1}{u}=\dfrac{1}{2}$ and $y=\dfrac{1}{v}=\dfrac{1}{3}$ .

Question 10

A boat goes $30$ km upstream and $44$ km downstream in $10$ hours. In $13$ hours it can go $40$ km upstream and $55$ km downstream. Find the speed of the boat in still water and the speed of the stream.

Accepted Answer

Let the boat's speed $=u$ km/h and the stream's speed $=v$ km/h. Put $p=\dfrac{1}{u-v}$ and $q=\dfrac{1}{u+v}$ .

$30p+44q=10\ \text{...(i)}$ , $\ 40p+55q=13\ \text{...(ii)}$ .

Multiply (i) by $4$ and (ii) by $3$ : $120p+176q=40$ and $120p+165q=39$ . Subtracting: $11q=1\Rightarrow q=\dfrac{1}{11}\Rightarrow u+v=11$ .

From (i): $30p+4=10\Rightarrow p=\dfrac{1}{5}\Rightarrow u-v=5$ .

Solving: $u=8,\ v=3$ . Boat $=8$ km/h, stream $=3$ km/h.

Question 11

Solve graphically the pair of equations $2x+y=6$ and $2x-y+2=0$. Shade the triangle formed by these two lines and the $x$-axis, and find its area.

Accepted Answer

For $2x+y=6$: points $(0,6)$ and $(3,0)$. For $2x-y+2=0$, i.e. $y=2x+2$: points $(0,2)$ and $(-1,0)$. The lines meet where $2x+y=6$ and $2x-y=-2$; adding, $4x=4\Rightarrow x=1,\ y=4$. So they intersect at $(1,4)$. The triangle formed with the $x$-axis has vertices $(-1,0),\ (3,0)$ and $(1,4)$, with base $=3-(-1)=4$ units and height $=4$ units. $$ ext{Area}= frac12 imes4 imes4=8\ ext{sq units.}$$ ```svg ```

Question 12

Places $A$ and $B$ are $100$ km apart on a highway. One car starts from $A$ and another from $B$ at the same time. If the cars travel in the same direction at different speeds, they meet in $5$ hours; if they travel towards each other, they meet in $1$ hour. Find the speeds of the two cars.

Accepted Answer

Let the speeds be $u$ km/h and $v$ km/h with $u>v$ .

Same direction (relative speed $u-v$ ): $100$ km covered in $5$ h $\Rightarrow 5(u-v)=100\Rightarrow u-v=20\ \text{...(i)}$ .

Opposite directions (relative speed $u+v$ ): $1(u+v)=100\Rightarrow u+v=100\ \text{...(ii)}$ .

Adding (i) and (ii): $2u=120\Rightarrow u=60$ ; then $v=40$ .

The speeds are $60$ km/h and $40$ km/h.

Question 13

A taxi service charges a **fixed** amount plus a rate per kilometre. Riya paid ₹$105$ for a $10$ km ride and ₹$155$ for a $15$ km ride. (i) Taking the fixed charge as ₹$x$ and the per-kilometre rate as ₹$y$, form the pair of linear equations. (ii) Find the fixed charge and the rate per kilometre. (iii) How much will a $25$ km ride cost?

Accepted Answer

**(i)** $x+10y=105\ 	ext{...(i)}$ and $x+15y=155\ 	ext{...(ii)}$. **(ii)** Subtracting (i) from (ii): $5y=50\Rightarrow y=10$. Then $x=105-10(10)=5$. So the fixed charge is ₹$5$ and the rate is ₹$10$ per km. **(iii)** Cost of a $25$ km ride $=x+25y=5+25(10)=$ ₹$255$.

Question 14

Are these Pair of Linear Equations in Two Variables important questions free?

Accepted Answer

Yes. All 13 CBSE Class 10 Maths important questions for Pair of Linear Equations in Two Variables are free, with full model answers and no login required.

Question 15

Do these Pair of Linear Equations in Two Variables questions follow the latest CBSE syllabus?

Accepted Answer

Yes — they are aligned to the NCERT 2026–27 syllabus for CBSE Class 10 Maths, so nothing here is outside the current course.

Question 16

How should I practise the Pair of Linear Equations in Two Variables important questions?

Accepted Answer

Attempt each question on paper first, then reveal the model answer to check your method — not just the final result. Re-do anything you got wrong the same day.

Question 17

What types of questions are covered for Pair of Linear Equations in Two Variables?

Accepted Answer

A full mix — multiple-choice questions, assertion–reason questions, very short answer questions, short answer questions, long answer questions, case-based questions — so every format in the CBSE paper is covered.

Question type	Count	Marks
MCQ	4	1
Assertion–Reason	1	1
Very Short	2	2
Short Answer	3	3
Long Answer	2	5
Case-based	1	4

Pair of Linear Equations in Two Variables — Important Questions

About Pair of Linear Equations in Two Variables

Key concepts & formulas

Get all 13 Pair of Linear Equations in Two Variables questions as a PDF

Important questions with answers

Multiple-choice questions (1 mark)

Assertion–Reason questions (1 mark)

Very short answer questions (2 marks)

Short answer questions (3 marks)

Long answer questions (5 marks)

Case-based questions (4 marks)

Frequently asked questions

Stuck on Pair of Linear Equations in Two Variables? Let the AI tutor help