Question

How do you find the general form of the line perpendicular to $3x + 5y - 8 = 0$ that passes through the point $\left( { - 8,1} \right)$?

Answer 1

Hint: This problem is regarding linear equations in two variables. Linear equation in two variables, explain the geometry of lines or the graph of two lines, plotted to solve the given equations. As we already know, the linear equation represents a straight line. If the two straight lines are perpendicular to each other then the product of their slopes is equal to -1, given by:
If slope of a line is ${m_1},$ and its perpendicular line slope is ${m_2}$, then :
$ \Rightarrow {m_1}{m_2} = - 1$

Complete step-by-step solution:
Given a linear equation which is given by: $3x + 5y - 8 = 0$, consider the equation below:
$ \Rightarrow 3x + 5y - 8 = 0$
$ \Rightarrow 5y = - 3x + 8$
$ \Rightarrow y = \dfrac{{ - 3}}{5}x + \dfrac{8}{5}$
This is in the form of a standard line equation which is given by $y = mx + c$, here $m$is the slope of the straight line and $c$ is the y-intercept.
So in the given straight line equation, $y = \dfrac{{ - 3}}{5}x + \dfrac{8}{5}$, the slope of the line is given by:
$ \Rightarrow m = \dfrac{{ - 3}}{5}$
So the straight line perpendicular to this line has the slope $\dfrac{{ - 1}}{m}$, which is given below:
$ \Rightarrow \dfrac{{ - 1}}{m} = \dfrac{5}{3}$
So the slope of the line perpendicular to the line $3x + 5y - 8 = 0$ is $\dfrac{5}{3}$.
The equation of the line perpendicular to the line $3x + 5y - 8 = 0$ is given by:
$ \Rightarrow y = {m_2}x + b$
$ \Rightarrow y = \dfrac{5}{3}x + b$
Here given that this line passes through the point (-8,1), hence substituting this point in the above line.
$ \Rightarrow 1 = \dfrac{5}{3}\left( { - 8} \right) + b$
$ \Rightarrow b = 1 + \dfrac{{40}}{3}$
$\therefore b = \dfrac{{43}}{3}$
Hence the equation of the line perpendicular to $3x + 5y - 8 = 0$ is given by:
$ \Rightarrow y = \dfrac{5}{3}x + \dfrac{{43}}{3}$
On simplifying the above equation, we get:
$ \Rightarrow 3y = 5x + 43$
$ \Rightarrow 5x - 3y - 43 = 0$

The general form of the line perpendicular to $3x + 5y - 8 = 0$ that passes through the point $\left( { - 8,1} \right)$ is $5x - 3y - 43 = 0$.

Note: Please note that an equation is said to be linear equation in two variables if it is written in the form of $ax + by + c = 0$, where $a,b$ and $c$ are real numbers and the coefficients of $x$ and $y$, i.e, $a$ and $b$ are not equal to zero.