Question

How do you find the slope that is perpendicular to the line $2x - 5y = 3?$

Answer 1

First find the slope of the tangent of the given line by finding the derivative of the line with respect to $x$ and then use the fact that product of slopes of a tangent and a normal or perpendicular equals negative one, to find the respective slope of the perpendicular to the given line.

Complete step by step solution:
In order to find the slope that is perpendicular to the given line $2x - 5y = 3$ we will first find slope of the line itself by calculating its derivative with respect to $x$
$\Rightarrow 2x - 5y = 3$
Differentiating both sides of the equation with respect to $x$, we will get

$$\Rightarrow \dfrac{{d(2x - 5y)}}{{dx}} = \dfrac{{d\left( 3 \right)}}{{dx}} \\\ \Rightarrow 2 - 5\dfrac{{dy}}{{dx}} = 0 \\\ \Rightarrow 5\dfrac{{dy}}{{dx}} = 2 \\\ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{2}{5} \\\$$

Now, we know that product of slopes of two perpendicular lines perpendicular to each other, is equals to $- 1$, that is if we consider the slope of a line ${m_1}$ and slope of another line which is perpendicular to the first line, to be ${m_2}$ then it can be written as
${m_1}{m_2} = - 1$
So, let us consider the slope of required perpendicular to be $m$, then we can write,
$\Rightarrow \dfrac{2}{5} \times m = - 1 \\\ \Rightarrow m = - \dfrac{5}{2} \\\$
Therefore $- \dfrac{5}{2}$ is the required slope of the perpendicular to the given line.

Note: take care when calculating the slope of the given line by differentiating it, and also that differentiation of a constant equals zero. And also take care of the signs when finding the slope of the perpendicular, because there is a chance of forgetting a negative sign before one when we write the product of slope.