Negative Slope Lines: Definition & Examples

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  • 0:41 Negative Slope in the…
  • 2:30 Negative Slope in an…
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Lesson Transcript
Instructor: David Liano
In this lesson, you'll learn how to identify and calculate a negative slope. We will work with negative slope lines using general mathematical graphics and symbols, and we'll look at negative slope lines put to use.

Definition of Negative Slope Lines

A line with a negative slope is a line that is trending downward from left to right. In other words, the line's rise to run ratio is a negative value.

Negative slope line

In this image, the car is driving down a hill that has a negative slope. Notice that the hill is moving downward from left to right. We can think of the car as driving down to the bottom of a hill. Yes, it could drive back up the hill, but the hill still has a negative slope because of its downward direction from left to right.

When determining if a line has a negative or positive slope, read the line's path from left to right, just as you read text in a book.

Negative Slope in the X-Y Coordinate Plane

Let's first look at negative slope lines in the standard x-y coordinate plane. This image shows the graph of two lines that are sloping downward from left to right. Therefore, they have negative slopes.

Negative slope lines
negative slope lines

The actual value of a slope is the line's rise to run ratio:

Slope = rise / run = (change in y) / (change in x)

Let's calculate the slope of the red line starting at coordinate (0, 6) and moving to (3, 3). The rise is three units down, so -3. The run is three units right, so +3. Therefore, the slope is -3/3. We can simplify this ratio to -1.

When calculating the rise of a line's slope, down is always negative and up is always positive. When calculating the run of a line's slope, right is always positive and left is always negative. We could find the slope by starting at coordinate (3, 3) instead. Then the rise would be +3 (three units up) and the run would be -3 (three units to the left); however, our slope is still -1. We can choose any two points on the line and will come up with the same slope. Notice that the green line is steeper. Therefore, its slope will be a larger negative number. Its actual slope is -2.

We also can calculate slope using the slope formula of:

slope formula

Let's find the slope of the red line again using this formula:

m = (3-6) / (3-0) = -3/3 = -1

Negative Slope in an Algebraic Equation

A linear equation can be shown in the slope-intercept form of:

y = mx + b

The letter m is the slope of the line, and the letter b is the y-intercept of the line.

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