Question

Given the line $2x - 3y - 5 = 0$, how do you find the slope of a line that is perpendicular to this line?

Answer 1

In the given problem, we are required to find the slope of a line which is perpendicular to the line whose equation is given to us. We can easily tell the slope of a line written in slope intercept form. So, we first have to write the given line in slope intercept form and then find the slope of the straight line given. For converting the line to slope intercept form, we need to have knowledge of algebraic methods like transposition rule.Then, we will find the slope of the perpendicular line using the concept that the product of slopes of perpendicular lines is always $- 1$.

Complete step by step answer:
Firstly, we are required to find the slope of the straight line $2x - 3y - 5 = 0$ .
Writing in standard form of equation of a straight line $ax + by + c = 0$ , we get,
$2x - 3y - 5 = 0$
Now shifting all terms except terms of y on other side, we get,
$\Rightarrow 3y = 2x - 5$
Finding the value of y by taking 3 to other side, we get,
$\Rightarrow y = \dfrac{{2x - 5}}{3}$

Further simplifying the equation and doing the calculations, we get,
$\Rightarrow y = \dfrac{2}{3}x - \dfrac{5}{3}$
On comparing with slope intercept form of straight line $y = mx + c$ where slope is given by ‘m’.Thus the slope of the straight line $2x - 3y - 5 = 0$ is $\dfrac{2}{3}$. Now, let the slope of the perpendicular line be x. We know that the product of slopes of two straight lines that are perpendicular to each other is $- 1$. So, we have,
Then, $\left( {\dfrac{2}{3} \times x} \right) = - 1$.
Finding value of x in the above equation, we get,
$\therefore x = - \dfrac{3}{2}$

So, the slope of the perpendicular line is $- \dfrac{3}{2}$.

Note: We can find the slope of a line by expressing it in point and slope form as well as slope and intercept form. We can also apply a direct formula for calculating the slope of a line: Slope of line$= - \left( {\dfrac{{{\text{Coefficient of x}}}}{{{\text{Coefficient of y}}}}} \right)$ when the equation of straight line is written in standard form $ax + by + c = 0$ . So, the slope of given line $2x - 3y - 5 = 0$ is $- \left( {\dfrac{2}{{ - 3}}} \right) = \dfrac{2}{3}$