NCERT Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables Exercise 3.2

NCERT Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables Exercise 3.2 are provided to support students in their Class 10 board exam preparation. These solutions are prepared by subject expert faculty to help students clearly understand the graphical method of solving linear equations.

Exercise 3.2 focuses on solving a pair of linear equations graphically. In this exercise, students learn how to draw graphs of two linear equations on the Cartesian plane and determine their point of intersection, which represents the solution of the given equations. The exercise also helps students understand different cases such as intersecting lines, parallel lines, and coincident lines.

NCERT Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables Exercise 3.2

Pair of Linear Equations in Two Variables

Medium

Q.

$\begin{array}{l}\text{On comparing the ratios }\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}\text{, }\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}\text{,\hspace{0.17em}\hspace{0.17em}}\frac{{\mathrm{c}}_1}{{\mathrm{c}}_2}\text{, find out whether}\\ \text{the following pair of linear equations are consistent,}\\ \text{or inconsistent.}\\ \\ \text{(i) }3\mathrm{x}+2\mathrm{y}=5;2\mathrm{x}-3\mathrm{y}=7\\ \text{(ii) }2\mathrm{x}-3\mathrm{y}=8;4\mathrm{x}-6\mathrm{y}=9\\ \text{(iii) }\frac{3}{2}\mathrm{x}+\frac{5}{3}\mathrm{y}=7;9\mathrm{x}-10\mathrm{y}=14\\ \text{(iv) \hspace{0.17em}}5\mathrm{x}-3\mathrm{y}=11;-10\mathrm{x}+6\mathrm{y}=-22\\ \text{\hspace{0.17em}(v) }\frac{4}{3}\mathrm{x}+2\mathrm{y}=8;2\mathrm{x}+3\mathrm{y}=12\end{array}$

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Pair of Linear Equations in Two Variables

Medium

Q.

$\begin{array}{l}\text{Which of the following pairs of linear equations are\hspace{0.33em}consistent/inconsistent?}\\ \text{If consistent, obtain the\hspace{0.33em}solution graphically:}\\ \text{(i) \hspace{0.33em}\hspace{0.33em}x\hspace{0.33em}+\hspace{0.33em}y\hspace{0.33em}=\hspace{0.33em}5,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}2x\hspace{0.33em}+\hspace{0.33em}2y\hspace{0.33em}=\hspace{0.33em}10 \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\\ \text{(ii)\hspace{0.33em}\hspace{0.33em}x\hspace{0.33em}–\hspace{0.33em}y\hspace{0.33em}=\hspace{0.33em}8,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}3x\hspace{0.33em}–\hspace{0.33em}3y\hspace{0.33em}=\hspace{0.33em}16}\\ \text{(iii)\hspace{0.33em}2x\hspace{0.33em}+\hspace{0.33em}y\hspace{0.33em}}–\text{6\hspace{0.33em}=\hspace{0.33em}0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}4x\hspace{0.33em}–\hspace{0.33em}2y\hspace{0.33em}}–\text{\hspace{0.33em}4\hspace{0.33em}=\hspace{0.33em}0}\\ \text{(iv)\hspace{0.33em}2x\hspace{0.33em}–\hspace{0.33em}2y\hspace{0.33em}–\hspace{0.33em}2\hspace{0.33em}=\hspace{0.17em}0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}4x\hspace{0.33em}–\hspace{0.33em}4y\hspace{0.33em}}–\text{\hspace{0.33em}5\hspace{0.33em}=\hspace{0.33em}0\hspace{0.17em}\hspace{0.17em}}\end{array}$

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Pair of Linear Equations in Two Variables

Medium

Q.

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.

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Pair of Linear Equations in Two Variables

Medium

Q.

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:
(i)   intersecting lines
(ii)  parallel lines
(iii) coincident lines

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Pair of Linear Equations in Two Variables

Medium

Q.

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.

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Pair of Linear Equations in Two Variables

Medium

Q.

Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
(i)   If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes ½ if we only add 1 to the denominator. What is the fraction?

(ii)  Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

(iv) Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹50 and  ₹100 notes only. Meena got 25 notes in all. Find how many notes of ₹ 50 and  ₹100 she received.

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

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The solutions are explained in a step-by-step and easy-to-understand manner, strictly following NCERT guidelines and the CBSE exam pattern, helping students gain confidence in solving graph-based questions.

Q1. 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 pair of linear equations are consistent or inconsistent. If consistent, obtain the solution graphically:

(i)

Equations:
$3x + 2y = 5$
$2x + 2y = 10$

Solution:

• $\frac{a_1}{a_2} = \frac{3}{2}$

• $\frac{b_1}{b_2} = \frac{2}{2} = 1$

• $\frac{c_1}{c_2} = \frac{5}{10} = \frac{1}{2}$

Since $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, these equations represent intersecting lines.
Answer: The lines intersect at a point, hence they are consistent.

(ii)

Equations:
$x - y = 8$
$3x - 3y = 16$

Solution:

• $\frac{a_1}{a_2} = \frac{1}{3}$

• $\frac{b_1}{b_2} = \frac{-1}{-3} = \frac{1}{3}$

• $\frac{c_1}{c_2} = \frac{8}{16} = \frac{1}{2}$

Since $\frac{a_1}{a_2} = \frac{b_1}{b_2}$, but $\frac{c_1}{c_2} \neq \frac{a_1}{a_2}$, these equations represent parallel lines.
Answer: The lines are parallel, hence the equations are inconsistent.

(iii)

Equations:
$\frac{3}{2}x + \frac{5}{3}y = 7$
$9x - 10y = 14$

Solution:

• $\frac{a_1}{a_2} = \frac{\frac{3}{2}}{9} = \frac{1}{6}$

• $\frac{b_1}{b_2} = \frac{\frac{5}{3}}{-10} = \frac{-5}{30} = \frac{-1}{6}$

• $\frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2}$

Since $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the lines are intersecting at a point.
Answer: The lines are consistent and intersect at a point.

(iv)

Equations:
$5x - 3y = 11$
$-10x + 6y = -22$

Solution:

• $\frac{a_1}{a_2} = \frac{5}{-10} = -\frac{1}{2}$

• $\frac{b_1}{b_2} = \frac{-3}{6} = -\frac{1}{2}$

• $\frac{c_1}{c_2} = \frac{11}{-22} = -\frac{1}{2}$

Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident, meaning they overlap and have infinitely many solutions.
Answer: The lines are coincident and consistent.

(v)

Equations:
$\frac{4}{3}x + 2y = 8$
$2x + 3y = 12$

Solution:

• $\frac{a_1}{a_2} = \frac{\frac{4}{3}}{2} = \frac{2}{3}$

• $\frac{b_1}{b_2} = \frac{2}{3}$

• $\frac{c_1}{c_2} = \frac{8}{12} = \frac{2}{3}$

Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident and have infinitely many solutions.
Answer: The lines are coincident and consistent.

Q2. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

(i)

Equations:
$x + y = 5$
$2x + 2y = 10$

Solution:
These are coincident lines.
Answer: Consistent and the lines overlap (infinitely many solutions).

(ii)

Equations:
$x - y = 8$
$3x - 3y = 16$

Solution:
These are parallel lines.
Answer: Inconsistent.

(iii)

Equations:
$2x + y - 6 = 0$
$4x - 2y - 4 = 0$

Solution:
These lines intersect at one point.
Answer: Consistent and intersecting.

(iv)

Equations:
$2x - 2y - 2 = 0$
$4x - 4y - 5 = 0$

Solution:
These are parallel lines.
Answer: Inconsistent.

Q3. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:

(i)

If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes ½ if we only add 1 to the denominator. What is the fraction?

Answer:
Let the fraction be $\frac{p}{q}$.
Equations:
$\frac{p+1}{q-1} = 1$
$\frac{p}{q+1} = \frac{1}{2}$
The solution is $\frac{3}{5}$.

(ii)

Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

Answer:
Let Nuri’s age be $x$ and Sonu’s age be $y$.
Equations:
$x - 5 = 3(y - 5)$
$x + 10 = 2(y + 10)$
Solution: Nuri’s age = 30 years, Sonu’s age = 10 years.

(iii)

The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

Answer:
Let the two-digit number be $10x + y$, where $x$ is the tens digit and $y$ is the ones digit.
Equations:
$x + y = 9$
$9(10x + y) = 2(10y + x)$
Solution: 45.

(iv)

Meena went to a bank to withdraw ₹2000. She asked the cashier to give her ₹50 and ₹100 notes only. Meena got 25 notes in all. Find how many notes of ₹50 and ₹100 she received.

Answer:
Let the number of ₹50 notes be $x$ and ₹100 notes be $y$.
Equations:
$x + y = 25$
$50x + 100y = 2000$
Solution: 20 ₹50 notes and 5 ₹100 notes.

(v)

A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹27 for a book kept for seven days, while Susy paid ₹21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

Answer:
Let the fixed charge be $x$ and the extra charge per day be $y$.
Equations:
$x + 4y = 27$
$x + 2y = 21$
Solution: Fixed charge = ₹15, Extra charge = ₹3/day.

FAQs: Class 10 Maths Chapter 3 – Exercise 3.2

Q1. What method is used in Exercise 3.2?
Answer:
Exercise 3.2 uses the graphical method to solve a pair of linear equations in two variables.

Q2. What does the point of intersection represent?
Answer:
The point of intersection of two lines represents the solution of the given pair of linear equations.

Q3. What are the possible cases of solutions in this exercise?
Answer:
There are three cases:

• Intersecting lines → one unique solution

• Parallel lines → no solution

• Coincident lines → infinitely many solutions

Q4. Is graph accuracy important in this exercise?
Answer:
Yes, accurate plotting of points and straight lines is essential to get the correct solution.

Q5. How do NCERT Solutions help in exams?
Answer:
These solutions provide clear graph-drawing steps, correct reasoning, and exam-oriented explanations, helping students score well in board examinations.