6.2 Slope of a Line

Leanne Thompson

6.2 Slope of a Line

When you graph linear equations, you may notice that some lines tilt up as they go from left to right, and other lines tilt down. Some lines are very steep, and other lines are flatter. In mathematics, the measure of the steepness of a line is called the slope of the line.

The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of pipes used for plumbing, and the steepness of a wheelchair ramp are all applications of slope. When you ski, bike, or jog on a hill, you may consider the slope of the hill because this will give you an idea of the difficulty level of that hill.

We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes, while the run measures the horizontal change.

Slope of a Line Ratio

Table 6.2.1
Slope of a Line Ratio
$m=\dfrac{\text{rise}}{\text{run}}$ Note: The letter m is used to denote slope.

Table 6.2.2
To use the slope of a line ratio:
Step 1: Locate two points on a line whose coordinates are integers. Step 2: Starting with the point on the left, sketch a right triangle, going from the first point to the second point. Step 3: Count the rise and the run on the legs of the triangle. Step 4: Take the ratio of rise to run to find the slope, $m=\dfrac{\text{rise}}{\text{run}}$ . Determine whether the slope is positive or negative by checking if the line is going up or down from left to right. Step 5: Reduce your ratio, if necessary.

Note that we “read” a line from left to right just like we read words in English. The line in the first figure is going up from left to right, so it has positive slope. The line in the second figure is going down from left to right, so it has negative slope.

The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled “Positive slope”. The line on the right is a diagonal line that drops from left to right. It is labeled “Negative slope”.

Example 1

Find the slope of the line below.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).

Table 6.2.3
Steps	Solution
Locate two points on the line whose coordinates are integers.
Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
Count the rise and the run on the legs of the triangle.
Take the ratio of rise to run to find the slope, $m=\dfrac{\text{rise}}{\text{run}}$ . Determine whether the slope is positive or negative by checking if the line is going up or down from left to right.	$m=\dfrac{4}{5}$ The slope is positive because the line goes up from left to right.
Reduce your ratio, if necessary.	The ratio cannot be reduced in this example.

Practice 1

Find the slope of each of the following linear relations.

a)

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 4. A line passes through the points (0, negative 1) and (4, 2).

b)

The graph shows the x y coordinate plane. The x-axis runs from negative 3 to 6 and the y-axis runs from negative 3 to 2. A line passes through the points (0, 1) and (5, negative 2).

c)

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1).

d)

The graph shows the x y coordinate plane. The x-axis runs from negative 4 to 2 and the y-axis runs from negative 6 to 2. A line passes through the points (negative 3, 4) and (1, 1).

Slope of a Horizontal Line

Example 2

Find the slope of $y=4$ .

Table 6.2.4
Steps	Solution
Draw a sketch of the graph, and locate two points on the line whose coordinates are integers.
Identify the rise and the run.	The rise is 0. The run is 3.
Take the ratio of rise to run to find the slope, $m=\dfrac{\text{rise}}{\text{run}}$ .	$\begin{array}{l}m=\dfrac{\text{rise}}{\text{run}}\\ m=\dfrac{0}{3}\\ m=0\end{array}$ The slope of the horizontal line $y=4$ is 0.

Practice 2

Find the slope of each line using the slope ratio.

a) $y=7$

b) $y=-5$

Slope of a Vertical Line

Example 3

Find the slope of $x=3$ .

Table 6.2.6
Steps	Solution
Draw a sketch of the graph, and locate two points on the line whose coordinates are integers.
Identify the rise and the run.	The rise is $2$ . The run is $0$ .
Take the ratio of rise to run to find the slope, $m=\dfrac{\text{rise}}{\text{run}}$ .	$\begin{array}{l} m=\dfrac{\text{rise}}{\text{run}}\\ m=\dfrac{2}{0}\end{array}$ The slope of the vertical line $x=3$ is undefined.

Practice 3

Find the slope of the lines using the slope ratio.

a) $x=8$

b) $x=-4$

To recap:

Table 6.2.7
Slope of a Horizontal Line	Slope of a Vertical Line
The slope of a horizontal line, $y=b$ , is 0.	The slope of a vertical line, $x=a$ , is undefined.

Graph a Line Given a Point and the Slope

Example 4

Graph the line that passes through the point (1, –1) whose slope is $m=\dfrac{3}{4}$ .

Table 6.2.8
Steps	Solution
Plot the given point.
Use the slope ratio to identify the rise and run.	$m=\dfrac{3}{4}$ $\dfrac{rise}{run}=\dfrac{3}{4}$ $rise=3$ $run=4$
Starting at the given point, count out the rise and run to mark the second point.
Connect the points with a line.

Practice 4

Draw the line that goes through the point (1, –3) and has a slope of $-\dfrac{3}{2}$ . Plot four points on the coordinate plane to help you draw the line.

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6 and a scale of 1.

Slope Word Problems

Practice 5

The “pitch” of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown below?

This figure shows a house with a sloped roof. The roof on one half of the building is labeled "pitch of the roof". There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled "rise equals 9 feet". There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled "run equals 18 feet".

Practice 6

The sewage pipes going away from your house to the street must slope downward $\dfrac{1}{4}$ -inch per foot in order to drain properly. What is the required slope of the pipe?

Homework

Find the slope of each of the following linear relations.

a)
b)

c)
d)

e)

f)
Find the slope of each line.

a) $y=3$ b) $y=1$ c) $x=4$ d) $x=2$

e) $y=-2$ f) $y=-3$ g) $x=-5$ h) $x=-4$
Graph a line with the given slope and point.

a) (1, –2); $m=\dfrac{3}{4}$

b) (2, 5); $m=-\dfrac{1}{3}$

c) (1, 5); m = –3

d) (–1, –4); $m=\dfrac{4}{3}$

e) y-intercept = 3; $m=-\dfrac{2}{5}$

f) x-intercept = –2; $m=\dfrac{3}{4}$
The rules for wheelchair ramps require a maximum 1-inch rise for a 12-inch run.
a) How long must the ramp be to accommodate a 24-inch rise to the door?
b) Draw a diagram of the ramp.
A local road has a grade of 6%. The grade of a road is its slope expressed as a percentage. Find the slope of the road as a fraction, and then simplify. What rise and run would reflect this slope or grade?
A hiker climbs a mountain. The elevation increases by 300 metres for every 2 kilometres of horizontal distance. What is the slope of the hike?
The elevation of a road decreases by 200 centimetres for every 20 metres of horizontal distance. What is the slope of the road?
A local road rises 2 feet for every 50 feet of highway.
a) What is the slope of the highway?
b) The grade of a highway is its slope expressed as a percentage. What is the grade of the highway?
What does the sign of the slope tell you about a line?
Why is the slope of a vertical line “undefined”?
How does the graph of a line with slope $m=\dfrac{1}{2}$ differ from the graph of a line with slope $m=2$ ?
The slope of a roof can be determined by measuring the change in height compared to the horizontal distance. If the roof rises feet in height for every $36$ feet of horizontal distance, and the slope of the roof is $\dfrac{1}{3}$ , what is the value of y?

Answers

1.

a) $\dfrac{2}{5}$

b) $-\dfrac{4}{3}$

c) $-3$

d) $\dfrac{3}{2}$

e) $-\dfrac{1}{3}$

f) $\dfrac{1}{4}$

2.

a) 0	b) 0	c) undefined	d) undefined
e) 0	f) 0	g) undefined	h) undefined

3.

a)

The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, negative 2) and (5, 1).

b)

The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (2, 5) and (5, 4).

c)

The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, 5) and (2, 2).

d)

The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 1, negative 4) and intercepts the x-axis at (2, 0).

e)

The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 3) and passes through the point (5, 1).

f)

The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 3).

4.

a) 288 inches or 24 feet
b)
Right triangle with a rise of 24 inches and a run of 288 inches.

5. $\dfrac{3}{50}$
rise = 3
run = 50

6. $\dfrac{3}{20}$

7. $-\dfrac{1}{10}$

8. a) $\dfrac{1}{25}$ b) $4%$

9. When the slope is a positive number, the line goes up from left to right. When the slope is a negative number, the line goes down from left to right.

10. A vertical line has 0 run, and since division by 0 is undefined, the slope is undefined.

11. A graph with a slope of $m=\dfrac{1}{2}$ has a rise of 1 and a run of 2, while a graph with a slope of $m=2$ has a rise of 2 and a run of 1.

12. 12 feet

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Marecek, L., & Honeycutt Mathis, A. (2020). Intermediate algebra 2e. OpenStax. https://collection.bccampus.ca/textbooks/key-concepts-of-intermediate-level-math-bccampus-204/, licensed under CC BY 4.0

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

License

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a) $y=3$	b) $y=1$	c) $x=4$	d) $x=2$
e) $y=-2$	f) $y=-3$	g) $x=-5$	h) $x=-4$