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negative slope — Definition, Formula & Examples

Negative slope is the slope of a line that falls (goes downward) as you move from left to right across the graph. Whenever the slope value mm is less than zero, the line has a negative slope.

A line has a negative slope when the ratio of the vertical change to the horizontal change between any two distinct points on the line is less than zero, i.e., m=y2−y1x2−x1<0m = \frac{y_2 - y_1}{x_2 - x_1} < 0. This occurs when yy decreases as xx increases, or equivalently when yy increases as xx decreases.

Key Formula

m=y2−y1x2−x1<0m = \frac{y_2 - y_1}{x_2 - x_1} < 0
Where:
  • mm = Slope of the line
  • (x1,y1)(x_1, y_1) = First point on the line
  • (x2,y2)(x_2, y_2) = Second point on the line

How It Works

To check whether a line has a negative slope, pick any two points on the line and compute m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. If the result is a negative number, the line slants downward from left to right. The steeper the downward slant, the more negative the slope value. For example, a slope of −5-5 drops much faster than a slope of −1-1. In real life, negative slopes describe situations where one quantity decreases as another increases — like the amount of battery charge decreasing over time.

Worked Example

Problem: Find the slope of the line through the points (1, 8) and (5, 2). Is it negative?
Step 1: Label the two points. Let (x1,y1)=(1,8)(x_1, y_1) = (1, 8) and (x2,y2)=(5,2)(x_2, y_2) = (5, 2).
Step 2: Substitute into the slope formula.
m=2−85−1=−64m = \frac{2 - 8}{5 - 1} = \frac{-6}{4}
Step 3: Simplify the fraction.
m=−32m = -\frac{3}{2}
Step 4: Since −32<0-\frac{3}{2} < 0, the slope is negative, meaning the line falls from left to right.
Answer: The slope is −32-\frac{3}{2}, which is negative. The line drops 3 units for every 2 units you move to the right.

Another Example

Problem: A line passes through (−2, 3) and (4, 3). Does it have a negative slope?
Step 1: Compute the slope using the formula.
m=3−34−(−2)=06=0m = \frac{3 - 3}{4 - (-2)} = \frac{0}{6} = 0
Step 2: A slope of 0 is not negative. This is a horizontal line — it neither rises nor falls.
Answer: No. The slope is 0 (a horizontal line), not negative.

Visualization

Why It Matters

Recognizing negative slopes is essential in 8th-grade algebra and pre-algebra courses whenever you graph linear equations or interpret real-world data. Scientists and economists rely on negative slopes to describe declining trends, such as a car losing value over time or a population shrinking each year. Mastering this concept also prepares you for perpendicular-slope problems, since the negative reciprocal of a positive slope is always negative (and vice versa).

Common Mistakes

Mistake: Subtracting the coordinates in mismatched order, like computing y2−y1x1−x2\frac{y_2 - y_1}{x_1 - x_2}, which flips the sign.
Correction: Always subtract in the same order: if you use y2−y1y_2 - y_1 in the numerator, you must use x2−x1x_2 - x_1 (not x1−x2x_1 - x_2) in the denominator.
Mistake: Thinking a negative slope means the line is "flat" or horizontal.
Correction: A horizontal line has a slope of exactly 0, not a negative slope. A negative slope means the line actively slants downward from left to right.

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