Class 10 Mathematics

Chapter 3 Solutions — Pair of Linear Equations in Two Variables

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Solutions

Overview

Step-by-step NCERT solutions for Pair of Linear Equations in Two Variables (Chapter 3, CBSE Class 10 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Pair of Linear Equations in Two Variables textbook chapter.

Solved

What these solutions cover

All 18 questions in Pair of Linear Equations in Two Variables are solved in the PDF. Here's what's inside, exercise by exercise:

Graphical Representation and Consistency

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.
- (ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one…
2.
On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
- (i) 5x - 4y + 8 = 0, 7x + 6y - 9 = 0
- (ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0
- (iii) 6x - 3y + 10 = 0, 2x - y + 9 = 0
3.
On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether the following pair of linear equations are consistent, or inconsistent.
- (i) 3x + 2y = 5; 2x - 3y = 7
- (ii) 2x - 3y = 8; 4x - 6y = 9
- (iii) (3/2)x + (5/3)y = 7; 9x - 10y = 14
- (iv) 5x - 3y = 11; -10x + 6y = -22

Solving by Substitution Method

1.
Solve the following pair of linear equations by the substitution method.
- (i) x + y = 14, x - y = 4
- (ii) s - t = 3, s/3 + t/2 = 6
- (iii) 3x - y = 3, 9x - 3y = 9
Solve 2x + 3y = 11 and 2x - 4y = -24 and hence find the value of 'm' for which y = mx + 3.
Form the pair of linear equations for the following problem and find their solution by substitution method. The difference between two numbers is 26 and one number is three times the other. Find them.
Solve the following pair of linear equations by the substitution method: 0.2x + 0.3y = 1.3 0.4x + 0.5y = 2.3
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 by substitution method.
5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and one pen.
The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

Solving by Elimination and Cross-Multiplication

1.
Solve the following pair of linear equations by the elimination method and the substitution method:
- (i) x + y = 5 and 2x - 3y = 4
- (ii) 3x + 4y = 10 and 2x - 2y = 2
- (iii) 3x - 5y - 4 = 0 and 9x = 2y + 7
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.
Form the pair of linear equations for the following problem and find their solution by elimination method. The sum of the digits of a two-digit number is 9. The number obtained by reversing the digits exceeds the original number by 27. Find the number.
Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her Rs 50 and Rs 100 notes only. Meena got 25 notes in all. Find how many notes of Rs 50 and Rs 100 she received.
A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for a book kept for five days. Find the fixed charge and the charge for each extra day.
6.
Solve the following pairs of equations by the cross-multiplication method:
- (i) x - 3y - 3 = 0, 3x - 9y - 2 = 0
- (ii) 2x + y = 5, 3x + 2y = 8
- (iii) 3x - 5y = 20, 6x - 10y = 40
- (iv) x - 3y - 7 = 0, 3x - 3y - 15 = 0
For which values of a and b does the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a - b)x + (a + b)y = 3a + b - 2 (ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1 (2k - 1)x + (k - 1)y = 2k + 1
8.
Solve the following pair of equations:
- (i) a/x - b/y = 0; ab²/x + a²b/y = a² + b², where x ≠ 0, y ≠ 0
- (ii) 2(ax - by) + (a + 4b) = 0; 2(bx + ay) + (b - 4a) = 0

Sample

Worked examples

3 questions from Pair of Linear Equations in Two Variables, solved step by step.

Graphical Representation and Consistency

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x - 4y + 8 = 0, 7x + 6y - 9 = 0 (ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0 (iii) 6x - 3y + 10 = 0, 2x - y + 9 = 0

(i) 5x - 4y + 8 = 0 and 7x + 6y - 9 = 0

a₁/a₂ = 5/7, b₁/b₂ = -4/6 = -2/3

Since a₁/a₂ ≠ b₁/b₂ (5/7 ≠ -2/3)

∴ The lines intersect at a point. (Consistent)

(ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0

a₁/a₂ = 9/18 = 1/2, b₁/b₂ = 3/6 = 1/2, c₁/c₂ = 12/24 = 1/2

Since a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2

∴ The lines are coincident. (Dependent, consistent — infinitely many solutions)

(iii) 6x - 3y + 10 = 0 and 2x - y + 9 = 0

a₁/a₂ = 6/2 = 3, b₁/b₂ = -3/-1 = 3, c₁/c₂ = 10/9

Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (3 = 3 ≠ 10/9)

∴ The lines are parallel. (Inconsistent — no solution)

Answer: (i) Intersecting (ii) Coincident (iii) Parallel

Solving by Substitution Method

Form the pair of linear equations for the following problem and find their solution by substitution method. The difference between two numbers is 26 and one number is three times the other. Find them.

Let the two numbers be x and y, where x > y.

x - y = 26 ... (1)

x = 3y ... (2)

Substituting (2) in (1):

3y - y = 26

2y = 26 ⇒ y = 13

x = 3(13) = 39

∴ The two numbers are 39 and 13.

Verification: 39 - 13 = 26 ✓ and 39 = 3 × 13 ✓

Answer: The numbers are 39 and 13

Solving by Elimination and Cross-Multiplication

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.

Let length = x m and width = y m.

Half perimeter: x + y = 36 ... (1)

Length is 4m more than width: x - y = 4 ... (2)

Using elimination:

Adding (1) and (2): 2x = 40 ⇒ x = 20

From (1): 20 + y = 36 ⇒ y = 16

∴ Length = 20 m, Width = 16 m

Verification: 20 + 16 = 36 ✓ and 20 - 16 = 4 ✓

Answer: Length = 20 m, Width = 16 m

All 18 questions in Pair of Linear Equations in Two Variables are solved step by step in the Solutions PDF above.

Board papers

Pair of Linear Equations in Two Variables in CBSE board papers

Tested in 72 questions across 25 Class 10 board papers (2020–2026). Open a set to see exactly how this chapter is asked:

Keep solving

Chapter 3 Solutions — Pair of Linear Equations in Two Variables

Overview

What these solutions cover

Graphical Representation and Consistency

Solving by Substitution Method

Solving by Elimination and Cross-Multiplication

Worked examples

Form the pair of linear equations for the following problem and find their solution by substitution method. The difference between two numbers is 26 and one number is three times the other. Find them.

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.

Pair of Linear Equations in Two Variables in CBSE board papers

More solutions in Mathematics

More NCERT Solutions for Class 10