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
Comparing these equation with
a1x + b1y + c1 = 0
a2x + b2y + c2= 0
We get
a1 = 5, b1 = -4, and c1 = 8
a2 =7, b2 = 6 and c2 = -9
a1/a2 = 5/7,
b1/b2 = -4/6 and
c1/c2 = 8/-9
Hence, a1/a2 ≠ b1/b2
Therefore, both are intersecting lines at one point.
(ii) 9x + 3y + 12 = 0
18x + 6y + 24 = 0
Comparing these equations with
a1x + b1y + c1 = 0
a2x + b2y + c2= 0
We get
a1 = 9, b1 = 3, and c1 = 12
a2 = 18, b2 = 6 and c2 = 24
a1/a2 = 9/18 = 1/2
b1/b2 = 3/6 = 1/2 and
c1/c2 = 12/24 = 1/2
Hence, a1/a2 = b1/b2 = c1/c2
Therefore, both lines are coincident
(iii) 6x – 3y + 10 = 0
2x – y + 9 = 0
Comparing these equations with
a1x + b1y + c1 = 0
a2x + b2y + c2= 0
We get
a1 = 6, b1 = -3, and c1 = 10
a2 = 2, b2 = -1 and c2 = 9
a1/a2 = 6/2 = 3/1
b1/b2 = -3/-1 = 3/1 and
c1/c2 = 12/24 = 1/2
Hence, a1/a2 = b1/b2 ≠ c1/c2
Therefore, both lines are parallel
