Parallel and Perpendicular Lines
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.
Parallel lines have the same slope. Perpendicular lines have slopes that are opposite reciprocals, unless one line is vertical and the other horizontal.
What you'll learn
- Identify parallel slopes
- Identify perpendicular slopes
- Write related line equations
Why it matters: Architecture drawings, road layouts, tile grids, and CAD designs depend on knowing whether lines stay aligned or meet at right angles. Same slopes keep lines parallel; opposite reciprocal slopes create perpendicular structure.
Worked example
Problem. What slope is perpendicular to 2/3?
- Flip 2/3 to get 3/2.
- Change the sign.
- The perpendicular slope is -3/2.
Answer: -3/2
Practice problems
1. What slope is parallel to -5?
Show solution
- Warm-up: First identify exactly what the question is asking: What slope is parallel to -5?
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Parallel slopes match.
- Check the result by substituting or estimating: the response should match -5 and make sense in the original problem.
Answer: -5
2. What slope is perpendicular to 4?
Show solution
- Core Practice: First identify exactly what the question is asking: What slope is perpendicular to 4?
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Use the opposite reciprocal.
- Check the result by substituting or estimating: the response should match -1/4 and make sense in the original problem.
Answer: -1/4
3. What slope is perpendicular to -2/7?
Show solution
- Challenge: First identify exactly what the question is asking: What slope is perpendicular to -2/7?
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Flip and change the sign.
- Check the result by substituting or estimating: the response should match 7/2 and make sense in the original problem.
Answer: 7/2
4. What slope is parallel to y = 2x + 1?
Show solution
- Parallel Lines: First identify exactly what the question is asking: What slope is parallel to y = 2x + 1?
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Parallel lines have the same slope.
- Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.
Answer: 2
5. What slope is perpendicular to 3/5?
Show solution
- Perpendicular Lines: First identify exactly what the question is asking: What slope is perpendicular to 3/5?
- For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
- Flip the fraction and change the sign.
- Check the result by substituting or estimating: the response should match -5/3 and make sense in the original problem.
Answer: -5/3
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