# Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using 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

**Solution:**

For any pair of linear equation

a₁ x + b₁ y + c₁ = 0

a₂ x + b₂ y + c₂ = 0

a) a₁/a₂ ≠ b₁/b₂ (Intersecting Lines)

b) a₁/a₂ = b₁/b₂ = c₁/c₂ (Coincident Lines)

c) a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (Parallel Lines)

(i) x - 3y - 3 = 0; 3x - 9 y - 2 = 0

a₁/a₂= 1/3

b₁/b₂= -3/-9 = 1/3

c₁/c₂= -3/-2 = 3/2

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Therefore, In the given problem the given sets of lines are parallel and no solution for these equations.

(ii) 2x + y = 5; 3x + 2y = 8

2x + y - 5 = 0

3x + 2 y - 8 = 0

a₁/a₂= 2/3

b₁/b₂= 1/2

c₁/c₂= -5/-8 = 5/8

a₁/a₂ ≠ b₁/b₂

Therefore, A unique solution for these equations.

By cross-multiplication method,

[x/(b₁c₂- b₂c₁) = y/(c₁a₂- c₂a₁) = 1/(a₁b₂ - a₂b₁)]

x/(-8 + 10) = y/(-15 + 16) = 1/(4 - 3)

x/2 = y/1 = 1

∴ x = 2 and y = 1

(iii) 3x - 5 y = 20; 6x - 10 y = 40

3x - 5y - 20 = 0

6x -10 y - 40 = 0

a₁/a₂= 3/6 = 1/2

b₁/b₂= 5/10 = 1/2

c₁/c₂= -20/-40 = 1/2

a₁/a₂ = b₁/b₂ = c₁/c₂

Therefore, Infinite solutions possible for these equations.

(iv) x - 3y - 7 = 0; 3x - 3y - 15 = 0

a₁/a₂_{ }= 1/3

b₁/b₂_{ }= -3/-3 = 1

c₁/c₂_{ }= -7/-15 = 7/15

a₁/a₂ ≠ b₁/b₂

Therefore, A unique solution for these equations.

By cross-multiplication method,

[x/(b₁c₂- b₂c₁) = y/(c₁a₂- c₂a₁)-= 1/(a₁b₂ - a₂b₁)]

[x/45 - 21) = y/-21 - (-15) = 1/-3 - (-9)]

x/24 = y/-6 = 1/6

x/24 = 1/6 and y/-6 = 1/6

x = 4 and y = - 1

∴ x = 4, y = -1

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 3

**Video Solution:**

## Which of the following pairs of linear equations has a unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using the cross multiplication method. (i) x - 3y - 3 = 0; 3x - 9 y - 2 = 0 (ii) 2x + y = 5; 3x + 2 y = 8 (iii) 3x - 5 y =20; 6x - 10 y = 40 (iv) x - 3y - 7 = 0; 3x - 3y - 15 = 0

NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.5 Question 1

The pair of equations that have unique solutions are ( 2x + y = 5; 3x + 2y = 8 ) and (x - 3y - 7 = 0; 3x - 3y - 15 = 0) no solutions are x - 3y - 3 = 0; 3x - 9 y - 2 = 0 infinitely many solutions are 3x - 5 y = 20; 6x - 10 y = 40

**☛ Related Questions:**

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