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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

Why it matters: Speed, cost per item, hourly earnings, temperature change, and revenue growth are all slope stories. The slope is the output units per input unit, so keeping the units attached makes the answer meaningful.

Worked example

Problem. Find the slope through (2, 5) and (6, 13).

  1. Change in y: 13 - 5 = 8.
  2. Change in x: 6 - 2 = 4.
  3. Slope = 8/4 = 2.

Answer: 2

Practice problems

1. Find the slope through (0, 3) and (4, 11).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the slope through (0, 3) and (4, 11).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. 8/4 = 2.
  4. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

2. Find the slope through (1, 9) and (5, 1).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the slope through (1, 9) and (5, 1).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Change in y is -8 and change in x is 4.
  4. Check the result by substituting or estimating: the response should match -2 and make sense in the original problem.

Answer: -2

3. A line with slope 0 is...

Choices: Horizontal · Vertical · Undefined · Curved

Show solution
  1. Challenge: First identify exactly what the question is asking: A line with slope 0 is...
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. No vertical change means horizontal.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Horizontal

4. Find the slope through (3, -2) and (9, 10).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the slope through (3, -2) and (9, 10).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Change in y is 12 and change in x is 6.
  4. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

5. A quantity rises 15 units when the input increases 5 units. What is the rate of change?

Show solution
  1. Rate of Change: First identify exactly what the question is asking: A quantity rises 15 units when the input increases 5 units. What is the rate of change?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. Rate of change is 15/5.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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