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 2 of 5

1. Q21 Exemplar · Exercise 3.1
   If $x=a$, $y=b$ is ​the solution ​of ​the equations $x-y=2$ and $x+y=4$, then ​the ​values of $a$ ​and $b$ ​are, respectively

   1. (a)3 and 5
   2. (b)5 and 3
   3. (c)3 and 1
   4. (d)$-1$ and $-3$
   Show answer

   (c) 3 and 1

   Step-by-step solution →
2. Q22 Exemplar · Exercise 3.1
   Aruna has ​only Re 1 ​and Rs 2 coins with ​her. If the total number of coins that she has is 50 and the amount of money with her ​is Rs 75, then ​the number of Re 1 and ​Rs ​2 coins ​are, respectively

   1. (a)35 and 15
   2. (b)35 and 20
   3. (c)15 and 35
   4. (d)25 and 25
   Show answer

   (d) 25 and 25

   Step-by-step solution →
3. Q23 Exemplar · Exercise 3.1
   The father's age is six times his son's age. ​Four ​years ​hence, the age of the father ​will be ​four times his son's age. The ​present ages, ​in years, of the son and the father are, respectively

   1. (a)4 and 24
   2. (b)5 and 30
   3. (c)6 and 36
   4. (d)3 and 24
   Show answer

   (c) 6 and 36

   Step-by-step solution →
4. Q24 Exemplar · Exercise 3.2
   Do the following ​pair ​of linear equations have no solution? Justify ​your answer.

   1. (i)$2x+4y=3$ ​and $12y+6x=6$
   2. (ii)$x=2y$ and $y=2x$
   3. (iii)$3x+y-3=0$ ​and $2x+\frac{2}{3}y=2$.

   1. i2x + 4y = 3 and 12y + 6x = 6
   2. iix = 2y and y = 2x
   3. iii$3x+y-3=0$ and $2x+(2/3)y=2$
   Show answer

   Only pair (i) has no solution (parallel lines). Pair (ii) has a unique solution $(0,0)$; pair (iii) has infinitely many solutions (coincident lines).

   Step-by-step solution →
5. Q25 Exemplar · Exercise 3.2
   Do ​the ​following equations represent a pair ​of ​coincident lines? Justify ​your ​answer.

   1. (i)$3x+\frac{1}{7}y=3$ ​and $7x+3y=7$
   2. (ii)$-2x-3y=1$ ​and $6y+4x=-2$
   3. (iii)$\frac{x}{2}+y+\frac{2}{5}=0$ and $4x+8y+\frac{5}{16}=0$
   Show answer

   Only pair (ii) represents coincident lines. Pair (i) gives intersecting lines and pair (iii) gives parallel (non-coincident) lines.

   Step-by-step solution →
6. Q26 Exemplar · Exercise 3.2
   Are the following pair ​of linear equations consistent? Justify your answer.

   1. (i)-3x - ​4y = 12 and 4y + 3x = 12.
   2. (ii)(3/5)x - y ​= 1/2 and ​(1/5)x - 3y = 1/6.
   3. (iii)​2ax + by = a ​and 4ax + ​2by - 2a = 0; a, b not equal to 0.
   4. (iv)​x + 3y = 11 and 2(2x + 6y) = 22.
   Show answer

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

   Step-by-step solution →
7. Q27 Exemplar · Exercise 3.2
   For the pair of equations $\lambda x+3y=-7$ and $2x+6y=14$ to have infinitely ​many solutions, the value ​of $\lambda$ should be 1. Is ​the ​statement ​true? Give reasons.
   Show answer

   False

   Step-by-step solution →
8. Q28 Exemplar · Exercise 3.2
   For all ​real values ​of $c$, the pair of equations $x-2y=8$ ​and $5x-10y=c$ have ​a unique solution. Justify ​whether it ​is ​true or false.
   Show answer

   False. Since $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{1}{5}$, the pair never has a unique solution for any $c$.

   Step-by-step solution →
9. Q29 Exemplar · Exercise 3.2
   The line represented ​by $x=7$ is parallel to the $x$-axis. ​Justify ​whether ​the statement ​is true or ​not.
   Show answer

   False

   Step-by-step solution →
10. Q30 Exemplar · Exercise 3.3
    For which value(s) ​of $\lambda$, ​do the pair of ​linear equations $\lambda x+y={\lambda }^2$ ​and $x+\lambda y=1$ have

    1. (i)no solution?
    2. (ii)​infinitely ​many solutions?
    3. (iii)a ​unique solution?
    Show answer

    (i) No solution: $\lambda =-1$. (ii) Infinitely many solutions: $\lambda =1$. (iii) Unique solution: $\lambda \ne \pm 1$.

    Step-by-step solution →
11. Q31 Exemplar · Exercise 3.3
    For ​which ​value(s) ​of $k$ ​will the pair of equations $kx+3y=k-3$ and $12x+ky=k$ ​have ​no ​solution?
    Show answer

    $k=-6$ is the only value giving no solution; $k=6$ gives infinitely many solutions and is rejected.

    Step-by-step solution →
12. Q32 Exemplar · Exercise 3.3
    For which ​values of $a$ and $b$, will the ​following pair ​of linear equations ​have ​infinitely ​many solutions? $x+2y=1$ ​and $(a-b)x+(a+b)y=a+b-2$.
    Show answer

    $a=3$ and $b=1$.

    Step-by-step solution →
13. Q33 Exemplar · Exercise 3.3
    Find the ​value(s) of $p$ in (i) to (iv) and $p$ ​and $q$ in (v) for the following pair of equations:

    1. (i)$3x-y-5=0$ ​and $6x-2y-p=0$, if the lines represented ​by these equations ​are ​parallel.
    2. (ii)$-x+py=1$ and $px-y=1$, if the pair of equations has ​no solution.
    3. (iii)$-3x+5y=7$ and $2px-3y=1$, if the lines represented by these equations are intersecting at a unique point.
    4. (iv)$2x+3y-5=0$ and $px-6y-8=0$, if the pair of ​equations has a unique solution.
    5. (v)$2x+3y=7$ and $2px+py=28-qy$, if the pair of equations have infinitely many solutions.
    Show answer

    (i) $p\ne 10$; (ii) $p=1$; (iii) $p\ne 9/10$; (iv) $p\ne -4$; (v) $p=4$, $q=8$

    Step-by-step solution →
14. Q34 Exemplar · Exercise 3.3
    Two ​straight paths ​are ​represented ​by ​the equations $x-3y=2$ and $-2x+6y=5$. Check whether the ​paths cross each ​other ​or not.
    Show answer

    The paths do not cross - the lines are parallel since $\frac{a_1}{a_2}=\frac{b_1}{b_2}=-\frac{1}{2}\ne \frac{2}{5}=\frac{c_1}{c_2}$.

    Step-by-step solution →
15. Q35 Exemplar · Exercise 3.3
    Write ​a ​pair of linear ​equations which has the unique ​solution $x=-1$, $y=3$. ​How ​many such pairs can you write?
    Show answer

    One valid pair: $x+y=2$ and $2x-y=-5$, with unique solution $(-1,3)$. Infinitely many such pairs can be written.

    Step-by-step solution →
16. Q36 Exemplar · Exercise 3.3
    If $2x+y=23$ ​and $4x-y=19$, ​find the ​values ​of $5y-2x$ and $\frac{y}{x}-2$.
    Show answer

    $x=7,y=9$; hence $5y-2x=31$ and $\frac{y}{x}-2=-\frac{5}{7}$.

    Step-by-step solution →
17. Q37 Exemplar · Exercise 3.3
    Find ​the ​values ​of $x$ ​and $y$ in ​the following ​rectangle ​[see Fig. 3.2].

    Show answer

    $x=1$ and $y=4$.

    Step-by-step solution →
18. Q38 Exemplar · Exercise 3.3
    Solve ​the ​following pairs ​of ​equations: ​

    1. (i)$x+y=3.3$, $\frac{0.6}{3x-2y}=-1$, $3x-2y\ne 0$.
    2. (ii)$\frac{x}{3}+\frac{y}{4}=4$, $\frac{5x}{6}-\frac{y}{8}=4$.
    3. (iii)$4x+\frac{6}{y}=15$, $6x-\frac{8}{y}=14$, $y\ne 0$. ​
    4. (iv)$\frac{1}{2x}-\frac{1}{y}=-1$, $\frac{1}{x}+\frac{1}{2y}=8$, $x,y\ne 0$.
    5. (v)$43x+67y=-24$, $67x+43y=24$.
    6. (vi)$\frac{x}{a}+\frac{y}{b}=a+b$, $\frac{x}{a^2}+\frac{y}{b^2}=2$, $a,b\ne 0$.
    7. (vii)$\frac{2xy}{x+y}=\frac{3}{2}$, $\frac{xy}{2x-y}=\frac{-3}{10}$, $x+y\ne 0,2x-y\ne 0$.
    Show answer

    (i) $x=1.2,y=2.1$; (ii) $x=6,y=8$; (iii) $x=3,y=2$; (iv) $x=\frac{1}{6},y=\frac{1}{4}$; (v) $x=1,y=-1$; (vi) $x=a^2,y=b^2$; (vii) $x=\frac{1}{2},y=-\frac{3}{2}$.

    Step-by-step solution →
19. Q39 Exemplar · Exercise 3.3
    Find the ​solution ​of the ​pair of ​equations $\frac{x}{10}+\frac{y}{5}-1=0$ and $\frac{x}{8}+\frac{y}{6}=15$. Hence, ​find $\lambda$, ​if $y=\lambda x+5$.
    Show answer

    $x=340,y=-165$; hence $\lambda =-\frac{1}{2}$.

    Step-by-step solution →
20. Q40 Exemplar · Exercise 3.3
    By the graphical ​method, find whether ​the ​following pair of equations ​are consistent or not. If consistent, ​solve them.

    1. (i)$3x+y+4=0$ ​and $6x-2y+4=0$ ​
    2. (ii)$x-2y=6$ ​and $3x-6y=0$
    3. (iii)$x+y=3$ and $3x+3y=9$.

    1. (i)$3x+y+4=0$; $6x-2y+4=0$
    2. (ii)x - 2y = 6; 3x - 6y = 0
    3. (iii)x + y = 3; 3x + 3y = 9
    Show answer

    (i) Consistent, intersecting, solution $x=-1,y=-1$. (ii) Inconsistent (parallel lines). (iii) Consistent, coincident - infinitely many solutions (every point on $x+y=3$).

    Step-by-step solution →