Do the following equations represent a pair of coincident lines? Justify your answer.

(i) and 7x + 3y = 7

(ii) - 2x - 3y = 1 and 6y + 4x = - 2

(iii)

Condition for coincident lines,

a₁/a₂ = b₁/b₂ = c₁/c₂;

(i) No.

Given pair of linear equations

and 7x + 3y = 7

Comparing with ax + by + c = 0;

Here, a₁ = 3, b₁ = 1/7, c₁ = - 3;

And a₂ = 7, b₂ = 3, c₂ = - 7;

a₁ /a₂ = 3/7

b₁ /b₂ = 1/21

c₁ /c₂ = - 3/ - 7 = 3/7

Here, a₁/a₂ b₁/b_2.

Hence, the given pair of linear equations has unique solution.

(ii) Yes, given pair of linear equations.

- 2x - 3y - 1 = 0 and 4x + 6y + 2 = 0;

Comparing with ax + by + c = 0;

Here, a₁ = - 2, b₁ = - 3, c₁ = - 1;

And a₂ = 4, b₂ = 6, c₂ = 2;

a₁ /a₂ = - 2/4 = - 1/2

b₁ /b₂ = - 3/6 = - 1/2

c₁ /c₂ = - 1/2

Here, a₁/a₂ = b₁/b₂ = c₁/c₂, i.e. coincident lines

Hence, the given pair of linear equations is coincident.

(ii) No, given pair of linear equations are

and = 0

Comparing with ax + by + c = 0;

Here, a₁ = 1/2, b₁ = 1, c₁ = 2/5;

And a₂ = 4, b₂ = 8, c₂ = 5/16;

a₁ /a₂ = 1/8

b₁ /b₂ = 1/8

c₁ /c₂ = 32/25

Here, a₁/a₂ = b₁/b₂ c₁/c₂, i.e. parallel lines

Hence, the given pair of linear equations has no solution.