Find the equation of the line passing through the point of intersection of the lines `4x-7y-3=0\ a n d\ 2x-3y+1=0` that has equal intercept to axes.

Explanation

Let the equation of a line passing through the point of intersection of the lines `2x+3y+1=0` and `3x-5y-5=0` be
`(2x+3y+1)+lambda(3x-5y-5)=0`
`implies x(2+3lambda)+y(3-5lambda)=5lambda-1`………`(1)`
`implies (x(2+3lambda))/(5lambda-1)+(y(3-5lambda))/(5lambda-1)=1`
`implies (x)/(((5lambda-1)/(2+3lambda)))+(y)/(((5lambda-1)/(3-5lambda)))=1`
Therefore, `x`-intercept `=(5lambda-1)/(2+3lambda)` and `y`-intercept `=(5lambda-1)/(3-5lambda)`
Given that `x`-intercept `=y-` intercept
`implies (5lambda-1)/(2+3lambda)=(5lambda-1)/(3-5lambda)`
`implies 2+3lambda=3-5lambda`
`implies lambda=(1)/(8)`
Put `lambda=(1)/(8)` in eq. `(1)`
`x(2+(3)/(8))+y(3-(5)/(8))=(5)/(8)-1`
`19x+19y=-3`