Unit 3 · Lesson 2 of 5
Slope
You'll be able to: find the slope of a line from its graph or from two points, and tell positive, negative, zero, and undefined slopes apart.
Step 1 of 5Warm-up
Two quick questions from earlier lessons. Answer from memory first. Pulling it out of your head is what makes it stick.
Find the x-intercept of \(4x + 3y = 12\). Write it as a point.
Set \(y = 0\): \(4x = 12\), so \(x = 3\).
Answer: \((3, 0)\).
\(\dfrac{2}{3} - \dfrac{5}{6}\)
The LCD is 6: \(\dfrac{2}{3} = \dfrac{4}{6}\).
\(\dfrac{4}{6} - \dfrac{5}{6} = -\dfrac{1}{6}\).
Step 2 of 5The idea
The slope of a line tells you how steep it is. It is written \(m\).
Pick two points on the line. The rise is how far you go up or down between them. The run is how far you go across. Slope is the rise divided by the run.
Slope from two points \((x_1, y_1)\) and \((x_2, y_2)\): \[m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\]
Read it left to right. Positive slope: the line goes up. Negative slope: it goes down.
Zero slope: a horizontal line (the rise is 0). Undefined slope: a vertical line (the run is 0, and you can't divide by 0).
A bigger absolute value of \(m\) means a steeper line: slope 3 is steeper than slope \(\tfrac{1}{2}\).
Why this shows up on the placement test: slope is inside every line question: writing equations, parallel and perpendicular lines, and rate-of-change word problems.
Step 3 of 5Worked examples
Example 1: slope from a graph
- Pick two clear points on the line: \((-2, 3)\) and \((2, -3)\).
- Run: from \(x = -2\) to \(x = 2\) is 4 to the right, so the run is \(4\).
- Rise: from \(y = 3\) down to \(y = -3\) is 6 down, so the rise is \(-6\). Down counts as negative.
- \(m = \dfrac{-6}{4} = -\dfrac{3}{2}\). The line goes down from left to right, so a negative slope makes sense.
Example 2: slope from two points, plus the two special cases
- Points \((-1, 4)\) and \((3, -4)\). Top: \(y_2 - y_1 = -4 - 4 = -8\).
- Bottom: \(x_2 - x_1 = 3 - (-1) = 3 + 1 = 4\). Minus a negative becomes plus.
- \(m = \dfrac{-8}{4} = -2\).
- Points \((2, 5)\) and \((-3, 5)\): \(m = \dfrac{5 - 5}{-3 - 2} = \dfrac{0}{-5} = 0\). A horizontal line.
- Points \((4, 1)\) and \((4, -6)\): \(m = \dfrac{-6 - 1}{4 - 4} = \dfrac{-7}{0}\). You can't divide by 0, so the slope is undefined. A vertical line.
Step 4 of 5Practice
Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went.
Find the slope of the line in the graph.
From \((0, -1)\) to \((3, 1)\): run 3 to the right, rise 2 up.
\(m = \dfrac{2}{3}\).
Find the slope of the line through \((2, 3)\) and \((6, 11)\).
\(m = \dfrac{11 - 3}{6 - 2} = \dfrac{8}{4} = 2\).
Find the slope of the line through \((-3, 5)\) and \((3, 1)\).
Top: \(1 - 5 = -4\). Bottom: \(3 - (-3) = 6\).
\(m = \dfrac{-4}{6} = -\dfrac{2}{3}\).
Find the slope of the line through \((-4, -2)\) and \((-1, -8)\).
Top: \(-8 - (-2) = -8 + 2 = -6\). Bottom: \(-1 - (-4) = -1 + 4 = 3\).
\(m = \dfrac{-6}{3} = -2\).
Find the slope of the line through \((5, -2)\) and \((5, 7)\).
\(m = \dfrac{7 - (-2)}{5 - 5} = \dfrac{9}{0}\).
Dividing by 0 is not allowed, so the slope is undefined. Both points have \(x = 5\): this is the vertical line \(x = 5\).
What kind of slope does the line through \((-2, 6)\) and \((4, 6)\) have?
- zero
- undefined
- positive
- negative
\(m = \dfrac{6 - 6}{4 - (-2)} = \dfrac{0}{6} = 0\).
Both points have \(y = 6\), so the line is horizontal. The slope is zero (a).
A plumber charges $150 for a 2-hour job and $285 for a 5-hour job. Think of these as the points \((2, 150)\) and \((5, 285)\). Find the slope. (It is the extra cost per hour, in dollars.)
\(m = \dfrac{285 - 150}{5 - 2} = \dfrac{135}{3} = 45\).
Answer: \(45\), meaning $45 more for each extra hour.
The line through \((2, k)\) and \((5, 10)\) has slope \(3\). Find \(k\).
Set up the slope: \(\dfrac{10 - k}{5 - 2} = 3\), so \(\dfrac{10 - k}{3} = 3\).
Multiply both sides by 3: \(10 - k = 9\), so \(k = 1\).
Check: \(\dfrac{10 - 1}{3} = 3\). Answer: \(k = 1\).
Step 5 of 5Wrap-up
You can now find slope as rise over run, \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\), and know that horizontal lines have slope 0 and vertical lines have undefined slope.
Read or watch next: OpenStax Intermediate Algebra 2e, 3.2 Slope of a Line (free textbook, with more practice at the end of the section).
Also good: Khan Academy: Slope (short videos). Quick rules: Lines and systems cheat sheet.
Something unclear? Ask your teacher bot a follow-up question, for example: "Why is a vertical line's slope undefined instead of zero?" It knows this course and can explain things another way.