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:

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