What are Slope, Rise and Run?
To calculate slope, you will need to understand the concepts of rise and run.
Slope
The slope of a line is a measurement of how much the line is tilted.
The formula for slope is:
\(\frac{rise}{run}= \frac{y2 – y1}{x2 – x1} = m\)
Rise
In the slope formula above, “rise” means the change in the y coordinates.
Run
On the other hand, “run” means the change in the x coordinates.
Slope – Free exercises
Instructions: Calculate the slope of the line by determining the rise and run of the following sets of coordinates.
1) (3,4) and (2,10)
2) (7,2) and (4,2)
3) (2,4) and (4, –6)
4) (4,2) and (–3,5)
5) (4,–3) and (–2,0)
Slope – Answers to the Exercises
Exercise 1: (3,4) and (2,10)
\(\frac{rise}{run}= \frac{y2 – y1}{x2 – x1} = m\)
\(\frac{10 – 4}{2 – 3} = m\)
\(\frac{6}{–1} = m\)
–6 = m
Exercise 2: (7,2) and (4,2)
\(\frac{y2 – y1}{x2 – x1} = m\)
\(\frac{2 – 2}{4 – 7} = m\)
\(\frac{0}{–3} = 0 = m\)
So, the coordinates represent a horizontal line.
Exercise 3: (2,4) and (4, –6)
\(\frac{–6 – 4}{4 – 2} = m\)
\(\frac{–10}{2} = –5= m\)
Exercise 4: (4,2) and (–3,5)
\(\frac{5 – 2}{–3 – 4} = m\)
\(\frac{3}{–7} = –\frac{3}{7} = m\)
Exercise 5: (4,–3) and (–2,0)
\(\frac{0 – –3}{–2– 4} = m\)
\(\frac{3}{–6} = –\frac{1}{2} = m\)
Using the Slope Formula
Here is the formula again:
\(\frac{rise}{run}= \frac{y2 – y1}{x2 – x1} = m\)
y1 is the first y coordinate used in the calculation. y2 is the second y coordinate used in calculation.
In the same way, x1 is the first x coordinate used in the calculation, and x2 is the second x coordinate used in calculation.
m is the variable that is used for the slope of the line.
Slope – Avoiding Common Mistakes
Avoid confusing coordinate 1 with coordinate 2 when substituting values into the slope formula.
Also be sure that your y coordinates are on the top of the fraction and the x coordinates are on the bottom.
You will calculate the slope incorrectly if you mix up the coordinates, so it pays to be careful.
Calculating Slope with Rise and Run – Example
Consider a straight line that has the following coordinates: (2,3) and (6,8).
What is the slope of this line?
How to solve:
First, you need to sort out which values go into each position in the formula.
Remember that coordinates are given in sets such as (x,y).
x and y coordinates
So, the x coordinate is given first in each set of parentheses, followed by the y coordinate.
Here, we have the coordinates (2,3) and (6,8), so y2 = 8 and y1 = 3.
For the x coordinates, x2 = 6 and x1 = 2.
Put the values in the formula
Then, you need to plug the values into the formula to solve.
\(\frac{rise}{run}= \frac{y2 – y1}{x2 – x1} = m\)
\(\frac{rise}{run}= \frac{8 – 3}{6 – 2} = m\)
\(\frac{5}{4} = m\)
Slope – Other Facts
Parallel lines
Parallel lines have the same slope.
Horozontal lines
A horizontal line has a slope of 0.
Vertical lines
A vertical line has no slope. So, we say that the slope of a vertical line is undefined.
Positive slope
A line that points upwards to the right has a positive slope.
Negative Slope
A line that points backwards to the left has a negative slope.
Slope – Further Problems
We will look at x and y intercepts and the point-slope formula in other posts.
