How to Calculate the Slope of a Line
In mathematics, the slope of a line is the direction and the steepness of the line. Slope is often denoted by the letter m;
Slope is calculated by finding the ratio of the “vertical change” to the “horizontal change” between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient (“rise over run”), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative “rise”. The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.
- A line is increasing if it goes up from left to right. The slope is positive, i.e. .
- A line is decreasing if it goes down from left to right. The slope is negative, i.e. .
- If a line is horizontal the slope is zero.
- If a line is vertical the slope is undefined
The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances – where the earth’s curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.
In mathematical language, the slope m of the line is
1. What is the correct order of respective slopes for the lines above?
a. Positive, undefined, negative, positive
b. Negative, zero, undefined, positive
c. Undefined, zero, positive, negative
d. Zero, positive undefined, negative
2. What is the slope of the line?
3. What is the slope of the line above?
4. What is the slope of the line above?
5. With the data given above, what is the value of y1?
(x1, y1) = (-9, 6) & (x2, y2) = (18, -18)
Slope=(-18 – 6) / [18 – (-9)] = -24/27= -(8/9)