NCERT Class 10 Maths – Chapter 3
Pair of Linear Equations in Two Variables
Exercise 3.1 – Complete Solutions
Class: 10
Subject: Mathematics
Chapter: Pair of Linear Equations in Two Variables
Exercise: 3.1
Important Concepts Used
- Standard form: ax + by + c = 0
- Condition for intersection:
- a₁/a₂ ≠ b₁/b₂ → Intersecting lines (Unique solution)
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → Parallel lines (No solution)
- a₁/a₂ = b₁/b₂ = c₁/c₂ → Coincident lines (Infinite solutions)
Question 1
(i) 10 students took part. Girls are 4 more than boys.
Let number of boys = x
Number of girls = y
x + y = 10
y = x + 4
Substitute:
x + (x + 4) = 10
2x + 4 = 10
2x = 6
x = 3
y = 3 + 4 = 7
Boys = 3, Girls = 7
(ii) Cost of pencils and pens
Let cost of one pencil = x
Cost of one pen = y
5x + 7y = 50
7x + 5y = 46
Multiply first by 7:
35x + 49y = 350
Multiply second by 5:
35x + 25y = 230
Subtract:
24y = 120
y = 5
Substitute:
5x + 7(5) = 50
5x + 35 = 50
5x = 15
x = 3
Pencil = ₹3, Pen = ₹5
Question 2
(i) 5x − 4y + 8 = 0
7x + 6y − 9 = 0
a₁/a₂ = 5/7
b₁/b₂ = −4/6 = −2/3
Ratios unequal → Intersecting lines (Unique solution)
(ii) 9x + 3y + 12 = 0
18x + 6y + 24 = 0
9/18 = 3/6 = 12/24
All ratios equal → Coincident lines (Infinite solutions)
(iii) 6x − 3y + 10 = 0
2x − y + 9 = 0
6/2 = −3/−1 ≠ 10/9
Intersecting lines (Unique solution)
Question 3
(i) 3x + 2y = 5 ; 2x − 3y = 7
a₁/a₂ ≠ b₁/b₂ → Consistent (Unique solution)
(ii) 2x − 3y = 8 ; 4x − 6y = 9
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → Inconsistent
(iii) (3/2)x + (5/3)y = 7 ; 9x − 10y = 14
Ratios unequal → Consistent
(iv) 5x − 3y = 11 ; −10x + 6y = −22
All ratios equal → Infinite solutions
(v) (4/3)x + 2y = 8 ; 2x + 3y = 12
Ratios unequal → Consistent
Question 4
(i) x + y = 5 ; 2x + 2y = 10
Coincident lines → Infinite solutions
(ii) x − y = 8 ; 3x − 3y = 16
Parallel lines → No solution
(iii) 2x + y − 6 = 0 ; 4x − 2y − 4 = 0
Intersecting → Unique solution
(iv) 2x − 2y − 2 = 0 ; 4x − 4y − 5 = 0
Parallel → No solution
Question 5
Let width = x
Length = x + 4
Half perimeter = 36
So, x + (x + 4) = 36
2x + 4 = 36
2x = 32
x = 16
Length = 20
Width = 16 m, Length = 20 m
Question 6
Given equation: 2x + 3y − 8 = 0- Intersecting: 2x − y + 1 = 0
- Parallel: 4x + 6y + 5 = 0
- Coincident: 4x + 6y − 16 = 0
Question 7
Equations:
x − y + 1 = 0
3x + 2y − 12 = 0
Solve simultaneously:
x − y = −1
3x + 2y = 12
Multiply first by 2: 2x − 2y = −2
Add:
5x = 10
x = 2
Substitute:
2 − y = −1
y = 3
Intersection point = (2, 3)
Vertices of triangle: (2,3), (0,6), (−1,0)