Question

The family of lines passing through the point of intersection of the lines $x+3y-5=0$ and $4x-y+1=0$ is given by

Answer 1

$(x+3y-5) + k(4x-y+1) = 0$

Answer 2

The equation of a family of lines passing through the point of intersection of two lines $L_1 = 0$ and $L_2 = 0$ is given by $L_1 + kL_2 = 0$, where $k$ is an arbitrary constant.

Given the two lines: $L_1: x+3y-5=0$ $L_2: 4x-y+1=0$

Substituting these into the formula, we get: $(x+3y-5) + k(4x-y+1) = 0$

This is the required equation for the family of lines.