Algebra helps students build strong logical thinking and problem-solving skills, and Chapter 3: Pair of Linear Equations in Two Variables is one of the most important chapters in Class 10 Maths. This chapter focuses on forming linear equations from real-life situations, solving them graphically, and understanding the nature of solutions (unique, no solution, or infinitely many solutions) by comparing coefficients.
These NCERT Solutions for Class 10 Maths Chapter 3 include important CBSE board questions asked between 2018 and 2025. All solutions are explained step by step in simple language to help students understand concepts clearly, practise efficiently, and score well in board examinations.
NCERT Solutions for Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables
Q.
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.
Q.
Find c if the system of equations cx + 3y + (3 – c) = 0; 12x + cy – c = 0 has infinitely many solutions?
[CBSE - 2019]
Q.
For the pair of linear equations
kx=y+2 and
6x=2y+3, determine if there exists any real value of
k for which the system has infinitely many solutions. Justify your answer.
Q.
If
x=1 and
y=2 is a solution for the pair of linear equations
2x–3y+a=0 and
2x+3y–b=0, determine the relationship between the constants
a and
b.
Q.
A fraction becomes
31 when 2 is subtracted from the numerator and it becomes
21 when 1 is subtracted from the denominator. Find the fraction.
[CBSE - 2019]
Q.
A father's age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.
[CBSE - 2019]
Q.
Half of the difference between two numbers is 2. The sum of the greater number and twice the smaller number is 13. Find the numbers.
[CBSE - 2023]
Q.
Three years ago, Rashmi was thrice as old as Nazma. Ten years later, Rashmi will be twice as old as Nazma. How old are Rashmi and Nazma now?
[CBSE - 2024]
Q.
In Figure, ABCD is a rectangle. Find the values of x and y.
[CBSE - 2018]
Q.
Solve the following system of linear equations 7x – 2y = 5 and 8x + 7y = 15 and verify your answer.
[CBSE - 2024]
Q.
The coach of a cricket team buys 3 bats and 6 balls for ₹ 3900. Later, she buys another bat and 2 more balls of the same kind for ₹ 1300. Represent this situation algebraically and geometrically.
Q.
The value of k for which the system of equations x + y – 4 = 0 and 2x + ky = 3, has no solution, is
[CBSE - 2020]
Q.
The value of k for which the pair of equations kx = y + 2 and 6x = 2y + 3 has infinitely many solutions,
[CBSE - 2023]
Q.
In the given figure, graph of two linear equations are shown. The pair of these linear equations is :
[CBSE - 2024]
Q.
If x = 1 and y = 2 is a solution of the pair of linear equations 2x – 3y + a = 0 and 2x + 3y – b = 0, then:
[CBSE - 2025]
Q.
ABCD is a cyclic quadrilateral. Find its all angles.
Q.
Q.
Form the pair of linear equations in the following problems, and find their solutions graphically.
(i) 10 students of Class X took part in 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.
(ii) 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 46. Find the cost of one pencil and that of one pen.
Q.
The cost of 2 kg of apples and 1 kg of grapes on a day was found to be ₹ 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹ 300. Represent the situation algebraically and geometrically.
Q.
Find the value of k for which the system of equations
x+y–4=0 and
2x+ky=3 has no solution.
Class 10 Maths Chapter 3 Questions & Answers – Pair of Linear Equations in Two Variables
Important Board Questions (CBSE 2018–2025)
Q1. (Medium)
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” Represent this situation algebraically and graphically.
Solution:
Let A = Aftab’s present age and D = daughter’s present age.
Seven years ago:
A − 7 = 7(D − 7) …(1)
Three years from now:
A + 3 = 3(D + 3) …(2)
Solving (1) and (2):
From (1): A − 7 = 7D − 49 ⇒ A = 7D − 42
Put in (2): 7D − 42 + 3 = 3D + 9 ⇒ 7D − 39 = 3D + 9 ⇒ 4D = 48 ⇒ D = 12
Then A = 7(12) − 42 = 84 − 42 = 42
Answer: Aftab’s age = 42 years, Daughter’s age = 12 years.
Graph: Plot the two lines A − 7D + 42 = 0 and A − 3D − 6 = 0. They intersect at (D, A) = (12, 42).
Q2. (Medium)
The coach of a cricket team buys 3 bats and 6 balls for ₹ 3900. Later, she buys another bat and 2 more balls of the same kind for ₹ 1300. Represent this situation algebraically and geometrically.
Solution:
Let x = cost of one bat, y = cost of one ball.
3x + 6y = 3900 …(1)
x + 2y = 1300 …(2)
Multiply (2) by 3:
3x + 6y = 3900 …(3)
(1) and (3) are the same equation, so the lines are coincident.
Hence, the pair has infinitely many solutions.
Geometrical representation: Both equations represent the same line (coincident lines).
Q3. (Medium)
The cost of 2 kg of apples and 1 kg of grapes was ₹ 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹ 300. Represent the situation algebraically and geometrically.
Solution:
Let x = cost of 1 kg apples, y = cost of 1 kg grapes.
2x + y = 160 …(1)
4x + 2y = 300 …(2)
Divide (2) by 2:
2x + y = 150 …(3)
(1) and (3) have the same left side but different constants, so the lines are parallel.
Hence, the pair has no solution.
Q4. (Medium)
Form the pair of linear equations and find their solutions graphically.
(i) 10 students took part. Girls are 4 more than boys.
Solution:
Let boys = x, girls = y.
x + y = 10 …(1)
y = x + 4 …(2)
From (2) in (1): x + (x + 4) = 10 ⇒ 2x = 6 ⇒ x = 3
Then y = 3 + 4 = 7
Answer: Boys = 3, Girls = 7.
Graph: Plot x + y = 10 and y − x = 4; intersection gives (3, 7).
(ii) 5 pencils and 7 pens cost ₹50; 7 pencils and 5 pens cost ₹46.
Solution:
Let pencil cost = x, pen cost = y.
5x + 7y = 50 …(1)
7x + 5y = 46 …(2)
Solving (1) and (2): x = 3, y = 5
Answer: Pencil = ₹3, Pen = ₹5.
Graph: Plot both lines; intersection gives (3, 5).
Q5. (Medium)
On comparing a1/a2, b1/b2, c1/c2, find whether the lines intersect, are parallel, or coincident.
(i) 5x − 4y + 8 = 0 and 7x + 6y − 9 = 0
Solution:
a1/a2 = 5/7, b1/b2 = (−4)/6 = −2/3
Since 5/7 ≠ −2/3, lines intersect ⇒ unique solution.
(ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0
Solution:
9/18 = 3/6 = 12/24 = 1/2
So lines are coincident ⇒ infinitely many solutions.
(iii) 6x − 3y + 10 = 0 and 2x − y + 9 = 0
Solution:
6/2 = 3, (−3)/(−1) = 3, but 10/9 ≠ 3
So lines are parallel ⇒ no solution.
Q6. (Medium)
ABCD is a cyclic quadrilateral. Find its all angles.
Solution:
In a cyclic quadrilateral, opposite angles are supplementary:
∠A + ∠C = 180° and ∠B + ∠D = 180°.
So, if any one angle (or a pair of adjacent angles) is given, the remaining angles can be found using the above relations.
Note: Numerical values of all angles cannot be determined unless at least one angle value (or more information) is provided in the question/figure.
Q7. (Easy) [CBSE – 2025]
If x = 1 and y = 2 is a solution of 2x − 3y + a = 0 and 2x + 3y − b = 0, then find a and b.
Solution:
Put x = 1, y = 2 in first equation:
2(1) − 3(2) + a = 0 ⇒ 2 − 6 + a = 0 ⇒ a = 4
Put x = 1, y = 2 in second equation:
2(1) + 3(2) − b = 0 ⇒ 2 + 6 − b = 0 ⇒ b = 8
Q8. (Medium) [CBSE – 2024]
In the given figure, graph of two linear equations are shown. The pair of these linear equations is:
Solution:
To write the equations from the graph:
1) Identify two points on each line (intercepts are easiest).
2) Use those points to find the equation of each line (using slope/intercept form).
3) Match with the given options (if MCQ).
Note: Exact equations depend on the values shown in the given figure.
Q9. (Easy) [CBSE – 2023]
The value of k for which the pair of equations kx = y + 2 and 6x = 2y + 3 has infinitely many solutions.
Solution:
kx = y + 2 ⇒ kx − y − 2 = 0 ⇒ a1=k, b1=−1, c1=−2
6x = 2y + 3 ⇒ 6x − 2y − 3 = 0 ⇒ a2=6, b2=−2, c2=−3
For infinitely many solutions:
a1/a2 = b1/b2 = c1/c2
But b1/b2 = (−1)/(−2) = 1/2 and c1/c2 = (−2)/(−3) = 2/3 (not equal).
So no value of k can make the system have infinitely many solutions.
Q10. (Easy) [CBSE – 2020]
The value of k for which x + y − 4 = 0 and 2x + ky = 3 has no solution.
Solution:
x + y − 4 = 0 ⇒ a1=1, b1=1, c1=−4
2x + ky − 3 = 0 ⇒ a2=2, b2=k, c2=−3
For no solution:
a1/a2 = b1/b2 ≠ c1/c2
1/2 = 1/k ⇒ k = 2
And c ratio = (−4)/(−3)=4/3 ≠ 1/2, condition satisfied.
Hence, k = 2.
Q11. (Medium) [CBSE – 2024]
Solve 7x − 2y = 5 and 8x + 7y = 15 and verify your answer.
Solution:
7x − 2y = 5 …(1)
8x + 7y = 15 …(2)
Solving gives x = 1, y = 1.
Verification:
Put x=1,y=1 in (1): 7(1)−2(1)=7−2=5 ✔
Put x=1,y=1 in (2): 8(1)+7(1)=8+7=15 ✔
Q12. (Medium) [CBSE – 2019]
Find c if cx + 3y + (3 − c) = 0 and 12x + cy − c = 0 has infinitely many solutions.
Solution:
a1=c, b1=3, c1=3−c
a2=12, b2=c, c2=−c
For infinitely many solutions:
c/12 = 3/c = (3−c)/(−c)
From c/12 = 3/c ⇒ c² = 36 ⇒ c = 6 (c ≠ 0)
Hence, c = 6.
Q13. (Easy) [CBSE – 2018]
In the figure, ABCD is a rectangle. Find the values of x and y.
Solution:
In a rectangle: opposite sides are equal and all angles are 90°.
Use the given angle/linear expressions from the figure and form two equations in x and y, then solve them.
Note: Exact values of x and y depend on the expressions shown in the given figure.
Q14. (Medium) [CBSE – 2024]
Three years ago, Rashmi was thrice as old as Nazma. Ten years later, Rashmi will be twice as old as Nazma. Find their present ages.
Solution:
Let Rashmi’s age = R, Nazma’s age = N.
Three years ago:
R − 3 = 3(N − 3) …(1)
Ten years later:
R + 10 = 2(N + 10) …(2)
Solving gives N = 16, R = 42.
Answer: Rashmi = 42 years, Nazma = 16 years.
Q15. (Medium) [CBSE – 2023]
Half of the difference between two numbers is 2. The sum of the greater number and twice the smaller number is 13. Find the numbers.
Solution:
Let greater number = x, smaller number = y.
(x − y)/2 = 2 ⇒ x − y = 4 …(1)
x + 2y = 13 …(2)
From (1): x = y + 4
Put in (2): y + 4 + 2y = 13 ⇒ 3y = 9 ⇒ y = 3
Then x = 3 + 4 = 7
Answer: Numbers are 7 and 3.
Q16. (Medium) [CBSE – 2019]
A father's age is three times the sum of ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.
Solution:
Let children’s present ages be x and y, father’s age be F.
F = 3(x + y) …(1)
After 5 years:
F + 5 = 2[(x + 5) + (y + 5)] = 2(x + y + 10) = 2(x + y) + 20 …(2)
Put (1) into (2):
3(x + y) + 5 = 2(x + y) + 20 ⇒ x + y = 15
So F = 3(15) = 45
Answer: Father’s present age = 45 years.
Q17. (Medium) [CBSE – 2019]
A fraction becomes 1/3 when 2 is subtracted from the numerator and it becomes 1/2 when 1 is subtracted from the denominator. Find the fraction.
Solution:
Let the fraction be n/d.
(n − 2)/d = 1/3 ⇒ 3n − 6 = d …(1)
n/(d − 1) = 1/2 ⇒ 2n = d − 1 ⇒ d = 2n + 1 …(2)
From (1) and (2):
3n − 6 = 2n + 1 ⇒ n = 7
Then d = 2(7) + 1 = 15
Answer: Fraction = 7/15.
Q18. (Easy)
If x=1 and y=2 is a solution of 2x − 3y + a = 0 and 2x + 3y − b = 0, determine the relationship between a and b.
Solution:
From Q7: a = 4 and b = 8.
So, b = 2a.
Q19. (Easy)
For kx = y + 2 and 6x = 2y + 3, determine if there exists any real value of k for which the system has infinitely many solutions. Justify.
Solution:
From Q9, b1/b2 = 1/2 but c1/c2 = 2/3 (not equal).
Hence, no real value of k can make the system have infinitely many solutions.
Q20. (Easy)
Find the value of k for which x + y − 4 = 0 and 2x + ky = 3 has no solution.
Solution:
From Q10, k = 2.
NCERT Solutions for Class 10 Maths Chapter 3 – FAQs
Q1. Why is Chapter 3 Pair of Linear Equations in Two Variables important for boards?
This chapter is frequently asked in CBSE exams and is a strong scoring area. It tests forming equations from word problems, graphical solutions, and conditions for unique/no/infinite solutions.
Q2. Which topics are most important in this chapter?
- Forming linear equations from real-life situations
- Graphical method of solving equations
- Conditions for unique, no solution, and infinitely many solutions
- CBSE case-study / concept-based questions
Q3. How to quickly identify the nature of solutions?
Compare coefficients:
Unique solution: a1/a2 ≠ b1/b2
No solution: a1/a2 = b1/b2 ≠ c1/c2
Infinite solutions: a1/a2 = b1/b2 = c1/c2
Q4. Is graphical method important for CBSE exams?
Yes. CBSE often asks questions where students must represent equations graphically and interpret intersection/parallel/coincident lines.
Q5. How should students prepare this chapter for full marks?
Practice word problems regularly, revise coefficient comparison conditions, and solve previous year CBSE questions to improve speed and accuracy.