Question

Find the equation of the line through the intersection of the lines $2x+3y-4=0$ and $x-5y=7$ that has its $x-$intercept equal to $-4$.

Answer 1

Equation of family of lines through two lines $L_1$ and $L_2$ are $L_1+\lambda L_2=0$

$L_1=>2x+3y-4=0L_2=>x-5y-7=0L_1+\lambda L_2=0=>2x+3y-4+\lambda (x-5y-7)=0=>x(2+\lambda )+y(3-5\lambda )-4-7\lambda =0$

$x-$intercept $=\frac{4+7\lambda }{2+\lambda }$

Given $x-$intercept $=-4$

$\therefore \frac{4+7\lambda }{2+\lambda }=-4=>\lambda =\frac{-12}{11}$

Therefore, equation of line is $2x+3y-4-\frac{12}{11}(x-5y-7)=0=>10x+93y+40=0$