Slope and Rate of Change
A free College Algebra lesson from the “Graphs and Forms of Linear Equations” unit, with a worked example and practice problems including step-by-step solutions.
Slope is the average rate of change of a linear relationship. It compares how much the output changes for each one-unit change in the input. In graphs, it is rise over run; in tables, it is change in y divided by change in x; in contexts, it is a rate such as dollars per item or miles per hour. This matters because slope is the central language of linear modeling. When practicing, keep units attached and subtract coordinates in the same order. A common mistake is using y-values over x-values inconsistently or confusing slope with the y-intercept.
What you'll learn
- Find slope from two points
- Interpret slope as rate of change
- Connect slope to linear graphs
Worked example
Problem. Find the slope through (2, 5) and (6, 13).
- Change in y: 13 - 5 = 8.
- Change in x: 6 - 2 = 4.
- Slope = 8/4 = 2.
Answer: 2
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Find the slope through (0, 3) and (4, 11).
Show solution
- 8/4 = 2.
Answer: 2
2. Find the slope through (1, 9) and (5, 1).
Show solution
- Change in y is -8 and change in x is 4.
Answer: -2
3. A line with slope 0 is...
Choices: Horizontal · Vertical · Undefined · Curved
Show solution
- No vertical change means horizontal.
Answer: Horizontal
4. Find the slope through (3, -2) and (9, 10).
Show solution
- Change in y is 12 and change in x is 6.
Answer: 2
5. A quantity rises 15 units when the input increases 5 units. What is the rate of change?
Show solution
- Rate of change is 15/5.
Answer: 3
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