Question

How do you find the slope of the line $( - 2,2)$ $(5,3)$ ?

Answer 1

Hint: The slope of a line is the steepness of a line in a horizontal or vertical direction. The slope of a line can be calculated by taking the ratio of the change in vertical dimensions upon the change in horizontal dimensions.

Formula used: Slope of a line can be given as $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$.

Complete step-by-step solution:
As we know the slope can be calculated by taking the ratio of vertical change and horizontal change.
Vertical change is $\Delta y = {y_2} - {y_1}$ .
Horizontal change is $\Delta x = {x_2} - {x_1}$ .
Now, here we are given two points. Let’s suppose them as
Point $1$ = $( - 2,2)$ and Point $2$ = $(5,3)$ .
Now, as we know that the first co-ordinate is the X coordinate and the second coordinate is the Y co-ordinate.
For point $1$ , ${x_1} = - 2$ and ${y_1} = 2$ .
For point $2$ , ${x_2} = 5$ and ${y_2} = 3$ .
Now, the vertical change is $\Delta y = {y_2} - {y_1}$ .
Substituting the value of ${y_1}$ and ${y_2}$ .
$\Delta y = (3) - (2)$
$ \Rightarrow \Delta y = 1$
Now, the horizontal change is $\Delta x = {x_2} - {x_1}$ .
Substituting the value of ${x_1}$ and ${x_2}$ .
$\Delta x = (5) - ( - 2)$
Simplifying the equation,
$\Delta x = 5 + 2$
$ \Rightarrow \Delta x = 7$
Now, we have the values of vertical change and horizontal change.
Now, to find the slope of the line, we take the ratio of vertical change to the horizontal change as shown.
$m = \dfrac{{\Delta y}}{{\Delta x}}$
The value of $\Delta y$ and $\Delta x$ are
$\Delta y = 1$
$\Delta x = 7$
Substituting the values in the equation of slope, we get
$m = \dfrac{1}{7}$

Hence the slope of the line is $\dfrac{1}{7}$.

Note: To find the slope, we subtracted the first coordinate from the second. But, if we inverse it and subtract the second coordinate from the first, then also there will be no change and the answer will remain the same. Also, a common mistake here might be forgetting the minus(-) sign in the x coordinate, which might lead to a wrong answer.