Question

Find the equation of the line parallel to the line passing through the point of intersection of the lines x + 2y - 3 = 0 and 4x - y + 7 = 0.​

Answer 1

$\Large\bf{\underline{\underline{Question:-}}}$

Find the equation of the line parallel to the line passing through the point of intersection of the lines x + 2y - 3 = 0 and 4x - y + 7 = 0.

$\Large\bf{\underline{\underline{Solution:-}}}$

The equation of the line passing through the point of intersection of the lines x + 2y - 3 = 0 and 4x - y + 7 = 0 is x + 2y - 3 + $\beta$ (4x - y + 7) = 0 ...........1

where, $\beta$ is a constant.

$\implies$ (1 + 4 $\beta$)x + (2 - $\beta$)y + (-3 +7$\beta$) = 0

Since, eqn 1 is parallel to the given line,

5x + 4y - 20 = 0

$\Large\therefore\: - \frac{1 + 4\beta}{2 - \beta} = - \frac{5}{4}$

$\Large\implies \: 4 + 16 \beta = 10 - 5 \beta$

$\Large\implies \: 16 \beta + 5 \beta = 10 - 4$

$\Large\implies \: 21 \beta = 6$

$\Large\implies \beta = \frac{6}{21} = \frac{2}{7}$

From eqn 1 , we get

$\large \: x + 2y - 3 \frac{2}{7} (4x - y + 7) = 0$

$\large\implies \: 7x - 14y - 21 + 8x - 2y + 14 = 0$

$\large\implies \: 15x + 12y - 7 = 0 ,$

which is the required equation of the line.

Hence, the required equation of the line is

${\boxed{\: 15x \: + \: 12y \: - \: 7 = 0}}$

$\pink{Hope \: it \: helps}$

Answer 2

Your required solution is in the attachment above✌️

${\huge{\boxed{\tt{\color{red}{Answer}}}}}$

The required equation of the line is,

15x + 12y - 7 = 0

$\huge\red{\boxed{\green{\mathbb{\overbrace{\underbrace{\fcolorbox{pink}{aqua}{\underline{\red{Hope it helps}}}}}}}}}$