# Slope of a Line

The **slope is the slant of a line**. All diagonal lines have a slope. Horizontal lines have a slope of 0. Vertical lines have an undefined slope.

Lines that slant upward, towards Quadrant I in the coordinate plane (where x and y are both positive), have **positive slopes**.

Lines that slant downward, towards Quadrant 4 in the coordinate plane (where x is positive and y is negative), have** negative slopes**.

More specifically, the slope of a line is $\dfrac{\text{rise}}{\text{run}}$ or the vertical distance between two points on a line over the horizontal distance between those same two points. We often find slope using the graph of the line by finding two places where the line crosses grid corners on a coordinate plane (always easy to derive whole numbers from a coordinate plane than decimals or fractions). We then draw a right angle that joins the two points. The length of the vertical portion of the angle is written over the length of the horizontal portion of the angle for the slope.

In the cases to the right, the rise is 8 and the run is 7, so the slope of the line is $\dfrac{\text{rise}}{\text{run}}=\dfrac{8}{7}$

If you have coordinates (because they are given, or because you can get them from the graph), you can calculate slope mathematically with the formula $\text{slope}=\dfrac{y_2-y_1}{x_2-x_1}$

For example:

If you have a line, like the one to the right, which passes through (4,6) and (-3,-2), then you can use the slope formula:

$$\eqalign{\text{slope}=\dfrac{y_2-y_1}{x_2-x_1}= \dfrac{-2-6}{-3-4}=\dfrac{-8}{-7}=\dfrac{8}{7}}$$

**Notice that the slope comes out the same, whether you find it by graphing or by using the formula.**

**Also note**: if the rise and the run are either both negative or both positive, then the slope is positive. If only the rise or only the run is negative, then the slope is negative (this follows basic integer division rules: a positive divided by a positive is positive, a negative divided by a negative is positive, but a positive divided by a negative is negative and a negative divided by a positive is negative).

**You can also compare the slope of various lines. **

Most straight lines intersect once. They have different slopes.

**Lines that do not intersect are parallel.** They have the **same slope** (so, the $m$ in each of these line's equations will be the same!).

The equations of these two lines would be:

$y=1x+2$ (top line) and

$y=1x-1$ (bottom line).

Note that the slope of each line is 1.

**Lines that insect in right angles are perpendicular**. These lines have slopes that are **opposite reciprocals** of each other (so, if the slope of one line is $\dfrac{3}{4}$ then the slope of teh perpendicular line is $\dfrac{-4}{3}$).

The equations of these two lines would be:

$y=1x+2$ (positive line) and

$y=-1x+3$ (negative line).

Note that the slope of the first line is $\dfrac{1}{1}$ and the second line is $\dfrac{-1}{1}$.

#### Practice Problems:

## Slope of a Line

What is the slope in the following equations?

- $y=4x+2$
- $y=-3x-1$
- $y=\dfrac{1}{2}x+7$
- $2y=4x+12$
- $6x-y=5$
- $-3y-7=2x$

Find the slope of the line that passes through the following two points:

- (3,4) and (8,9)
- (-2,3) and (-4,7)
- (-6,-2) and (1,1)
- (1,2) and (-3, -4)
- (9,2) and the origin
- (3,3) and (3,7)

#### Answer Key: