Topic 4 of 8

Slope and graphing lines

Step 10 · Pre-algebra topics

Find the slope of a line from two points or a graph, read and write equations y = mx + b, and see the unit rate of a proportional relationship as a slope.

Slope as rise over run

Slope measures how steep a line is: how much it rises for each step to the right. Pick two points and divide the rise by the run.

m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}

From (1, 2) to (5, 8): run 4 and rise 6, so the slope is 6 ÷ 4 = 1.5.

Example: Slope from two points

Find the slope of the line through (2, 7) and (6, −1).

Rise: −1 − 7 = −8.
Run: 6 − 2 = 4.
Slope: −8 ÷ 4 = −2.

Answer−2

Slope-intercept form

y = mx + b is slope-intercept form. m is the slope, and b is the y-intercept: where the line crosses the y-axis.

y = 2x + 1: start at (0, 1) on the y-axis, then go 1 across and 2 up each time.

Example: Reading the equation

What are the slope and y-intercept of y = −3x + 5?

m = −3: down 3 for every 1 across.
b = 5: it crosses the y-axis at (0, 5).

AnswerSlope −3, y-intercept 5

Example: Writing an equation

A line has slope 4 and crosses the y-axis at (0, −2). Write its equation.

m = 4 and b = −2.

Answery = 4x − 2

Example: From two points

Write the equation of the line through (1, 5) and (3, 9).

Slope: rise 4 over run 2 is 2.
So y = 2x + b. Put in (1, 5): 5 = 2 + b, so b = 3.

Answery = 2x + 3

Proportional relationships and slope

A proportional relationship y = kx is a line through the origin. Its slope is the unit rate, k.

Example: Comparing two rates

Pump A fills 3 gallons a minute. Pump B's graph is a line through (0, 0) and (4, 10). Which pump is faster?

B's slope: 10 ÷ 4 = 2.5 gallons a minute.
A fills 3 gallons a minute.

AnswerPump A

Two slope triangles on y = 0.5x: 2 across and 1 up, and 6 across and 3 up. Both give 0.5.

Try it yourself

Work each one on paper first, then open the answer to check.

Find the slope of the line through (0, 1) and (4, 9).
Show answer
2
Find the slope of the line through (−2, 5) and (2, −3).
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−2
What are the slope and y-intercept of y = 0.5x − 4?
Show answer
Slope 0.5, y-intercept −4
Write the equation of the line with slope −1 that crosses the y-axis at 6.
Show answer
y = −x + 6
Write the equation of the line through (2, 3) and (4, 7).
Show answer
y = 2x − 1
A runner's distance is d = 6t (miles and hours). What is the slope, and what does it mean?
Show answer
6: she runs 6 miles each hour.