Class 10 Maths · Chapter 3

Pair of Linear Equations in Two Variables

81 Class 10 Maths questions on Pair of Linear Equations in Two Variables from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 1 of 5

1. Q1 NCERT · Exercise 3.1
   Form ​the pair of linear equations in the following problems, and find their solutions graphically: ​(i) $10$ students of Class X ​took part in a Mathematics quiz. If the number ​of girls is $4$ more than the number of boys, find the ​number of boys ​and ​girls who took part in the ​quiz.
   Show answer

   There are $3$ boys and $7$ girls (the lines $x+y=10$ and $y-x=4$ meet at $(3,7)$).

   Step-by-step solution →
2. Q2 NCERT · Exercise 3.1
   On ​comparing ​the ​ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, find ​out whether ​the lines representing ​the following pairs of linear equations intersect at a point, are parallel or coincident:

   1. (i)$5x-4y+8=0$ ​and $7x+6y-9=0$
   2. (ii)$9x+3y+12=0$ and $18x+6y+24=0$
   3. (iii)$6x-3y+10=0$ and $2x-y+9=0$
   Show answer

   (i) The lines intersect at a point; (ii) The lines are coincident; (iii) The lines are parallel.

   Step-by-step solution →
3. Q3 NCERT · Exercise 3.1
   On ​comparing ​the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, find out ​whether the ​following pairs of linear equations are consistent or ​inconsistent:

   1. (i)$3x+2y=5$ ​and $2x-3y=7$
   2. (ii)$2x-3y=8$ and $4x-6y=9$ ​
   3. (iii)$\frac{3}{2}x+\frac{5}{3}y=7$ and $9x-10y=14$
   4. (iv)$5x-3y=11$ and $-10x+6y=-22$ ​
   5. (v)$\frac{4}{3}x+2y=8$ and $2x+3y=12$
   Show answer

   (i) Consistent; (ii) Inconsistent; (iii) Consistent; (iv) Consistent (infinitely many solutions); (v) Consistent (infinitely many solutions).

   Step-by-step solution →
4. Q4 NCERT · Exercise 3.1
   Which of the ​following pairs ​of ​linear equations are consistent/inconsistent? ​If consistent, ​obtain ​the solution graphically:

   1. (i)$x+y=5$ and $2x+2y=10$
   2. (ii)$x-y=8$ ​and $3x-3y=16$
   3. (iii)$2x+y-6=0$ and $4x-2y-4=0$
   4. (iv)$2x-2y-2=0$ and $4x-4y-5=0$
   Show answer

   (i) Consistent — coincident lines, infinitely many solutions; (ii) Inconsistent — no solution; (iii) Consistent — unique solution $(2,2)$; (iv) Inconsistent — no solution.

   Step-by-step solution →
5. Q5 NCERT · Exercise 3.1
   Half the ​perimeter of a rectangular ​garden, whose length is $4$ m more ​than its width, is $36$ m. Find the ​dimensions ​of ​the garden.
   Show answer

   The garden is $20$ m long and $16$ m wide.

   Step-by-step solution →
6. Q6 NCERT · Exercise 3.1
   Given the ​linear equation $2x+3y-8=0$, ​write another linear equation ​in two variables ​such that the ​geometrical representation ​of ​the pair so ​formed is:

   1. (i)intersecting lines
   2. (ii)parallel lines
   3. (iii)coincident lines
   Show answer

   (i) Intersecting: $3x+2y-7=0$; (ii) Parallel: $4x+6y-5=0$; (iii) Coincident: $4x+6y-16=0$. (Any equation meeting the same ratio condition is acceptable.)

   Step-by-step solution →
7. Q7 NCERT · Exercise 3.1
   Draw the graphs of the equations $x-y+1=0$ ​and $3x+2y-12=0$. Determine the coordinates of the ​vertices of ​the triangle formed by ​these lines and the $x$-axis, ​and shade the ​triangular ​region.
   Show answer

   The vertices of the triangle are $A(-1,0)$, $B(2,3)$ and $C(4,0)$; the enclosed region is shaded in the graph.

   Step-by-step solution →
8. Q8 NCERT · Exercise 3.2
   Solve ​the following pair of linear ​equations ​by the substitution method: ​

   1. (i)$x+y=14$ ​and $x-y=4$
   2. (ii)$s-t=3$ and $\frac{s}{3}+\frac{t}{2}=6$ ​
   3. (iii)$3x-y=3$ and $9x-3y=9$
   4. (iv)$0.2x+0.3y=1.3$ ​and $0.4x+0.5y=2.3$
   5. (v)$\sqrt{2}x+\sqrt{3}y=0$ and $\sqrt{3}x-\sqrt{8}y=0$
   6. (vi)$\frac{3x}{2}-\frac{5y}{3}=-2$ and $\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$
   Show answer

   (i) $x=9,y=5$; (ii) $s=9,t=6$; (iii) Infinitely many solutions ($y=3x-3$); (iv) $x=2,y=3$; (v) $x=0,y=0$; (vi) $x=2,y=3$.

   Step-by-step solution →
9. Q9 NCERT · Exercise 3.2
   Solve $2x+3y=11$ and $2x-4y=-24$ ​and ​hence ​find the ​value of ​'$m$' ​for which $y=mx+3$.
   Show answer

   $x=-2,y=5$, and $m=-1$.

   Step-by-step solution →
10. Q10 NCERT · Exercise 3.2
    Form the pair of linear equations for the following problems ​and find their solution by the substitution method:

    1. (i)The difference between two numbers is $26$ and one number is three times the ​other. Find them.
    2. (ii)The larger of two supplementary ​angles exceeds the smaller by $18$ degrees. Find them.
    3. (iii)The coach of a cricket team buys $7$ bats and $6$ balls for Rs $3800$. Later, she buys $3$ bats and $5$ balls for Rs $1750$. Find the cost of each bat and each ​ball.
    4. (iv)The taxi ​charges in a city consist of a fixed charge together with the charge ​for the distance covered. For a distance of $10$ km, the charge paid is Rs $105$ and ​for a journey of $15$ km, the charge paid is Rs $155$. What are the fixed charges and the charge per km? How much does a person ​have to pay for travelling a distance of $25$ km?
    5. (v)A fraction becomes $\frac{9}{11}$ if $2$ is added to both the numerator and the denominator. If $3$ is added to both the numerator and the denominator it becomes $\frac{5}{6}$. Find the fraction.
    6. (vi)Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob's age was seven times that of his son. What are their present ages?
    Show answer

    (i) 39 and 13; (ii) $99^{\circ }$ and $81^{\circ }$; (iii) bat Rs 500, ball Rs 50; (iv) fixed Rs 5, Rs 10 per km, Rs 255 for 25 km; (v) $7/9$; (vi) Jacob is 40 and his son is 10.

    Step-by-step solution →
11. Q11 Exemplar · Exercise 3.1
    Graphically, ​the pair of equations $6x-3y+10=0$ ​and $2x-y+9=0$ represents ​two lines ​which are

    1. (a)intersecting at exactly one point.
    2. (b)intersecting at exactly two points.
    3. (c)coincident.
    4. (d)parallel.
    Show answer

    (d) parallel.

    Step-by-step solution →
12. Q12 Exemplar · Exercise 3.1
    The pair ​of ​equations $x+2y+5=0$ and $-3x-6y+1=0$ ​have

    1. (a)a unique solution
    2. (b)exactly two solutions
    3. (c)infinitely many solutions
    4. (d)no solution
    Show answer

    (d) no solution

    Step-by-step solution →
13. Q13 Exemplar · Exercise 3.1
    If a pair of linear ​equations is ​consistent, ​then ​the ​lines ​will be

    1. (a)parallel
    2. (b)always coincident
    3. (c)intersecting or coincident
    4. (d)always intersecting
    Show answer

    (c) intersecting or coincident

    Step-by-step solution →
14. Q14 Exemplar · Exercise 3.1
    The pair of ​equations $y=0$ ​and $y=-7$ ​has

    1. (a)one solution
    2. (b)two solutions
    3. (c)infinitely many solutions
    4. (d)no solution
    Show answer

    (d) no solution

    Step-by-step solution →
15. Q15 Exemplar · Exercise 3.1
    The pair of ​equations $x=a$ ​and $y=b$ ​graphically represents ​lines ​which are

    1. (a)parallel
    2. (b)intersecting at $(b,a)$
    3. (c)coincident
    4. (d)intersecting at $(a,b)$
    Show answer

    (d) intersecting at $(a,b)$

    Step-by-step solution →
16. Q16 Exemplar · Exercise 3.1
    For ​what ​value ​of $k$, do ​the ​equations $3x-y+8=0$ ​and $6x-ky=-16$ ​represent coincident lines?

    1. (a)$\frac{1}{2}$
    2. (b)$-\frac{1}{2}$
    3. (c)2
    4. (d)$-2$
    Show answer

    (c) 2

    Step-by-step solution →
17. Q17 Exemplar · Exercise 3.1
    If ​the lines given by $3x+2ky=2$ ​and $2x+5y+1=0$ are ​parallel, ​then the value ​of $k$ is

    1. (a)$\frac{-5}{4}$
    2. (b)$\frac{2}{5}$
    3. (c)$\frac{15}{4}$
    4. (d)$\frac{3}{2}$
    Show answer

    (c) $\frac{15}{4}$

    Step-by-step solution →
18. Q18 Exemplar · Exercise 3.1
    The value ​of $c$ for ​which the pair ​of ​equations $cx-y=2$ and $6x-2y=3$ will have ​infinitely many ​solutions is

    1. (a)$3$
    2. (b)$-3$
    3. (c)$-12$
    4. (d)no value
    Show answer

    (d) no value

    Step-by-step solution →
19. Q19 Exemplar · Exercise 3.1
    One equation ​of a pair ​of dependent linear ​equations ​is $-5x+7y=2$. ​The ​second ​equation can ​be

    1. (a)$10x+14y+4=0$
    2. (b)$-10x-14y+4=0$
    3. (c)$-10x+14y+4=0$
    4. (d)$10x-14y=-4$
    Show answer

    (d) $10x-14y=-4$

    Step-by-step solution →
20. Q20 Exemplar · Exercise 3.1
    A pair ​of linear ​equations ​which ​has a ​unique solution $x=2$, $y=-3$ ​is

    1. (a)$x+y=-1$ and $2x-3y=-5$
    2. (b)$2x+5y=-11$ and $4x+10y=-22$
    3. (c)$2x-y=1$ and $3x+2y=0$
    4. (d)$x-4y-14=0$ and $5x-y-13=0$
    Show answer

    (d) $x-4y-14=0$ and $5x-y-13=0$

    Step-by-step solution →