Question

How do you find the slope of the line passing through the points $\left( { - 2,5} \right)$,$\left( {6, - 3} \right)$?

Answer 1

Here, we will substitute the given points into the formula of the slope of the line. We will then simplify it further to get the required slope. The slope is defined as the ratio of change in the $y$ to the change in the $x$.

Complete Step by Step Solution:
We are given two points $\left( { - 2,5} \right)$, $\left( {6, - 3} \right)$. These two points are the points in a line.
Now, we will find the slope of the line passing through the points $\left( { - 2,5} \right)$ and $\left( {6, - 3} \right)$.
Slope of the line passing through two points is given by the formula $m = \dfrac{{\Delta y}}{{\Delta x}}$ i.e., where $\Delta y$ is a change in $y$ and $\Delta x$ is a change in $x$ i.e. $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ where $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ are the coordinates of the points respectively.
We have $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ as $\left( { - 2,5} \right)$ and $\left( {6, - 3} \right)$.
By substituting the given points in the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$, we get
$m = \dfrac{{ - 3 - 5}}{{6 - \left( { - 2} \right)}}$
$\Rightarrow m = \dfrac{{ - 3 - 5}}{{6 + 2}}$
Adding and subtracting the terms, we get
$\Rightarrow m = \dfrac{{ - 8}}{8}$
By simplifying the equation, we get
$\Rightarrow m = - 1$

Therefore, the slope of the line passing through the points $\left( { - 2,5} \right)$ and $\left( {6, - 3} \right)$ is $- 1$.

Note:
We know that Slope can be represented in the parametric form and in the point form. A point crossing the x-axis is called x-intercept and A point crossing the y-axis is called the y-intercept. The slope of a line is used to calculate the steepness of a line. We know that the horizontal line does not run vertically i.e., ${y_2} - {y_1}$ , so the slope is zero and also the vertical line does not run horizontally i.e.,${x_2} - {x_1}$ , the slope is undefined. These slopes are obtained by using the slope of a line using two points formula.