Find the equation of the line passing through the intersection of the lines `4x+7y-3=0 and 2x-3y+1=0` that has equal intercepts on the axes.

Explanation

Let equation of a line passing through the point of intersection of lines `4x+7y-3=0` and `2x-3y+1=0` is,
`(4x+7y-3)+lambda(2x-3y+1)=0`
`implies x(4+2lambda)+y(7-3lambda)=3-lambda`…….`(1)`
`implies(x)/((3-lambda)/(4+2lambda))+(y)/((3-lambda)/(7-3lambda))=1`
`:.x-"intercept"=(3-lambda)/(4+2lambda)`
and `y-"intercept"=(3-lambda)/(7-3lambda)`
Given that , `(3-lambda)/(4+2lambda)=(3-lambda)/(7-3lambda)`
`implies (3-lambda)(7-3lambda)=(3-lambda)(4+2lambda)`
`implies (3-lambda)(7-3lambda-4-2lambda)=0`
`implies (3-lambda)(3-5lambda)=0`
`implies lambda=0` or `lambda=(3)/(5)`
But `lambda=3` is not possible.
`:'` In this case, constant term `=0`
Now, put `lambda=(3)/(5)` in equation `(1)` ,
`x(4+(6)/(5))+y(7-(9)/(5))=3-(3)/(5)`
`(26x)/(5)+(26y)/(5)=(12)/(5)implies13x+13y=6`