Question

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂ , and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide:
(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

Answer 1

(i) 5x - 4y + 8 = 0,

7x + 6y - 9 = 0

A condition for the lines to be parallel is a1/a2 = b1/b2 ≠ c1/c2

but, 5/7 ≠ -4/6 ≠ 9/-9

Therefore the equations are consistent and intersect each other at a point.

Therefore, these equations coincide.

(ii) 9x + 3y + 12 = 0,

18x + 6y + 24 = 0

A condition for the lines to coincide is a1/a2 = b1/b2 = c1/c2

9/18 = 3/6 = 12/24

1/2 = 1/2 = 1/2

The linear equations coincide as one of the equation is a multiple of the other.

(iii) 6x - 3y + 10 = 0,

2x - y + 9 = 0

A condition for the lines to be parallel is a1/a2 = b1/b2 ≠ c1/c2

6/2 = -3/-1 ≠ 10/9

3 = 3 ≠ 10/9

Therefore, these equations are parallel.