Question

How do you find the slope from the pair of points $\left( -5,-4 \right)$ and $\left( 5,2 \right)$ ?

Answer 1

We are given 2 points $\left( -5,-4 \right)$ and $\left( 5,2 \right)$ , we are asked to find the slope of the line which will pass through these points, to answer this we start by learning about what does slope mean, what are the method we focus on finding slope using ratio of coordinates or using the angle that is formed with the horizontal axis, we also learn about slope intercept form to get the slope.

Complete step-by-step solution:
We are given a pair of points which are $\left( -5,-4 \right)$ and $\left( 5,2 \right)$ , we have to find the slope using those pairs of points.
To answer this we first learn about the definition and the ways to find the slope.
Now, the slope of any line is the angle made by the line with the positive x-axis.
We generally find the slope by finding the ratio of rise and run.
Rise means movement of the function along the y-axis, while run refers to the movement along x-axis.
So, one way is slope $\text{=}\dfrac{\text{rise}}{\text{run}}$.
Another way is to find the term of the angle made by the line with the x-axis.
So, slope $=\tan \theta$ .

Slope is denoted as ‘m’, so $m=\tan \theta$ or $m=\dfrac{\text{rise}}{\text{run}}$ .
Other ways to find the slope is to use the equation given to us.
Generally the equation of line in standard form is given as $ax+by+c=0$ .
We can convert this equation to slope intercept form given as –
$y=mx+c$ .
Where ‘m’ is a slope, ‘c’ is the y-intercept. So, we can find slope and intercept from here.
Now we have two points $\left( -5,-4 \right)$ and $\left( 5,2 \right)$ we start by considering these points as $\left( {{x}_{1}},{{y}_{1}} \right)=\left( -5,-4 \right)$ and other point as $\left( {{x}_{2}},{{y}_{2}} \right)=\left( 5,2 \right)$ .
Now, we will use $\text{slope}=\dfrac{\text{rise}}{\text{run}}$ .
Run is the difference in x-interval. So, $\text{run}={{x}_{2}}-{{x}_{1}}$ .
As ${{x}_{2}}=5$ and ${{x}_{1}}=-5$ .
So,
$\begin{aligned} & \text{run}={{x}_{2}}-{{x}_{1}}=5-\left( -5 \right) \\\ & =10 \\\ \end{aligned}$ .
So, we get run = 10.
Similarly, we will find rise.
The rise is the difference of y-interval.
$\text{rise}={{y}_{2}}-{{y}_{1}}$ .
As we have ${{y}_{2}}=2$ and ${{y}_{1}}=-4$ .
So, we get –
$\begin{aligned} & \text{rise}=2+4 \\\ & =6 \\\ \end{aligned}$
Now we have rise = 6 and run = 10.
So,
$\text{slope}=\dfrac{\text{rise}}{\text{run}}$
$=\dfrac{6}{10}$ .
By simplifying, we get –
$\text{slope}=\dfrac{3}{5}$ .

Note: We can also directly apply the value
$\text{slope}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ .
Since, we have ${{y}_{2}}=2,{{y}_{1}}=-4,{{x}_{2}}=5,{{x}_{1}}=-5$ .
So, $\text{slope}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ .
$=\dfrac{2-\left( -4 \right)}{5-\left( -5 \right)}=\dfrac{6}{10}$ .
So, $\text{slope}=\dfrac{3}{5}$ .
If we have the equation of line $ax+by+c=0$ then here the short way to find the slope is the ratio of ‘-a’ and ‘b’ $\text{slope}=\dfrac{-a}{b}$ .