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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.

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