We can solve examples of the slope of a line using the slope formula. This formula is derived by dividing the change in y by change in x. The slope defines the inclination of the line. A positive slope indicates that the line grows from left to right, whereas a negative slope indicates that the line decreases from left to right.
Here, we will review the slope formula and use it to solve some practice problems.
Review of the formula for the slope of a line
The formula for the slope of a line is obtained by dividing the change in y of two points that are located on the line by the change in x of the points. Therefore, if we have two points $latex A = (x_{1}, y_{1})$ and $latex B = (x_{2}, y_{2})$ that are located on the same line, the slope formula is:
| Formula for the slope $latex m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ |
Slope of a line – Examples with answers
The following examples are solved using the formula for the slope of a line. Try to solve the problems yourself before looking at the solution.
EXAMPLE 1
What is the slope of a line that has the points (2, 1) and (4, 5)?
Solution
We have the following coordinates:
- $latex (x_{1}, y_{1})=(2, 1)$
- $latex (x_{2}, y_{2})=(4, 5)$
We use the slope formula with the given coordinates:
$latex m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$latex m=\frac{5-1}{4-2}$
$latex m=\frac{4}{2}$
$latex m=2$
The slope of the line is 2.
EXAMPLE 2
We have the points (3, 2) and (6, 3) that are part of a line. What is the slope?
Solution
We have the following values:
- $latex (x_{1}, y_{1})=(3, 2)$
- $latex (x_{2}, y_{2})=(6, 3)$
Using the slope formula with these values, we have:
$latex m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$latex m=\frac{3-2}{6-3}$
$latex m=\frac{1}{3}$
The slope of the line is $latex \frac{1}{3}$.
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EXAMPLE 3
Determine the slope of a line containing the points (-1, 3) and (6, -4).
Solution
We write the values as follows:
- $latex (x_{1}, y_{1})=(-1, 3)$
- $latex (x_{2}, y_{2})=(6, -4)$
Using these values in the formula, we have:
$latex m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$latex m=\frac{-4-3}{6-(-1)}$
$latex m=\frac{-7}{7}$
$latex m=-1$
The slope of the line is $latex -1$.
EXAMPLE 4
If a line has the points (-2, 1) and (6, -3), what is its slope?
Solution
We have the coordinates:
- $latex (x_{1}, y_{1})=(-2, 1)$
- $latex (x_{2}, y_{2})=(6, -3)$
We apply the slope formula with the given coordinates:
$latex m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$latex m=\frac{-3-1}{6-(-2)}$
$latex m=\frac{-4}{8}$
$latex m=-\frac{1}{2}$
The slope of the line is $latex -\frac{1}{2}$.
Slope of a line – Practice problems
The following problems can be solved by applying what you have learned about the slope of a line. In case you need help with this, you can look at the solved examples above.
See also
Interested in learning more about distance, midpoint, and slope on the plane? Take a look at these pages:
