Given:

lines given by (2 + k)x + (1 + k)y = 5 + 7k

To prove:

The straight lines given by (2 + k)x + (1 + k)y = 5 + 7k for different values of k pass through a fixed point

Explanation:

The given straight line (2 + k)x + (1 + k)y = 5 + 7k can be written in the following way:

2x + y − 5 + k (x + y − 7) = 0

This line is of the form L₁ + kL₂ = 0, which passes through the intersection of the lines
L₁ = 0 and L₂ = 0, i.e. 2x + y − 5 = 0 and x + y − 7 = 0.

Solving 2x + y − 5 = 0 and x + y − 7 = 0, we get (− 2, 9), which is the fixed point.

Hence proved.