Question

Gradient of a line perpendicular to the line $3x-2y=5$ is
A. $-\dfrac{2}{3}$
B. $\dfrac{2}{3}$
C. $-\dfrac{3}{2}$
D. $-\dfrac{5}{2}$

Answer 1

First we will convert the given equation into a general slope-intercept form of a line. The general equation of slope-intercept form of a line is given as $y=mx+c$ where, m is the slope of the line and c is the y-intercept of the line. Then we will use the property that the product of slopes of two lines perpendicular to each other is $-1$ to get the desired answer.

Complete step by step answer:
We have been given an equation of a line $3x-2y=5$.
We have to find the gradient of a line perpendicular to the line $3x-2y=5$.
We know that the slope-intercept form of a line is given by the equation $y=mx+c$ where, m is the slope of the line and c is the y-intercept of the line. Y-intercept of the line is the point where a line crosses the Y-axis.
Now, let us convert the given equation in the general form. Then we will get
$\begin{aligned} & \Rightarrow 3x-2y=5 \\\ & \Rightarrow 3x-5=2y \\\ & \Rightarrow y=\dfrac{3}{2}x-\dfrac{5}{2} \\\ \end{aligned}$
Now, comparing the given equation with the general equation we will get
$\Rightarrow m=\dfrac{3}{2}$ and $\Rightarrow c=-\dfrac{5}{2}$
Now, we know that the product of slopes of two lines perpendicular to each other is $-1$.
So we will get that
$\begin{aligned} & \Rightarrow m'\times \dfrac{3}{2}=-1 \\\ & \Rightarrow m'=\dfrac{-2}{3} \\\ \end{aligned}$
Hence we get the gradient of a line perpendicular to the line $3x-2y=5$ is $\dfrac{-2}{3}$.

So, the correct answer is “Option A”.

Note: The point to be noted is that while calculating the slope of the line the coefficient of y must be 1. Alternatively we can find the slope and intercept of the given equation by using the graphing method. For this we draw the graph of a given equation which is a straight line and then we can find the slope and intercept of the obtained line.