Given:

2x + 3y + 1 = 0 and 3x – 5y – 5 = 0

To find:

The equation of the straight line passing through the point of intersection of 2x + 3y + 1 = 0 and 3x – 5y – 5 = 0 and equally inclined to the axes.

Explanation:

The equation of the straight line passing through the points of intersection of 2x + 3y + 1 = 0 and 3x − 5y − 5 = 0 is

2x + 3y + 1 + λ(3x − 5y − 5) = 0

⇒ (2 + 3λ)x + (3 − 5λ)y + 1 − 5λ = 0
⇒ y

The required line is equally inclined to the axes. So, the slope of the required line is either 1 or − 1.

∴ and

⇒ -2 – 3λ = 3 – 5λ and 2 + 3λ = 3 – 5λ

⇒ λ and

Substituting the values of λ in (2 + 3λ)x + (3 − 5λ)y + 1 − 5λ = 0, we get the equations of the required lines.

⇒ 19x – 19y – 23 = 0 and 19x + 19y + 3 = 0

Hence, required equation is 19x – 19y – 23 = 0 and 19x + 19y + 3 = 0