Question

5x $-$ 2y + 4 = 0 is a linear equation. Write another equation in two variables such that the geometrical representation of the pair so formed is as follows
(a) parallel lines
(b) intersecting lines
(c) overlapping (coincident) lines.

Answer 1

(a) Two equations are equations of parallel lines if it satisfies the following:

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

An equation of a line parallel to 5x $-$ 2y + 4 = 0 is 10x $-$ 4y + 5 = 0.

(b) Two equations are equations of intersecting lines if it satisfies the following:

$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

An equation of a line intersecting 5x $-$ 2y + 4 = 0 is 4x $-$ 2y + 5 = 0.

(c) Two equations are equations of overlapping (coincident) lines if it satisfies the following:

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

An equation of a line overlapping 5x $-$ 2y + 4 = 0 is 15x $-$ 6y + 12 = 0.