Point-Slope Form
Learn how to write and use the point-slope form of a linear equation when you know a point and the slope.
Definition
- is the slope of the line
- is a specific point on the line
- represents any other point on the line
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Worked Examples
Write the equation of a line with slope that passes through the point .
Identify the values
Slope , Point → , ,
Write point-slope form
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Substitute values
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Answer:
Common Mistakes
Forgetting to change the sign when or is negative
Why it's wrong: In , if , then , not .
Correct: Always write the subtraction first, then simplify:
Using the wrong point for
Why it's wrong: When given two points, students sometimes mix coordinates from different points.
Correct: Pick ONE point completely. Both coordinates must come from the same point.
Distributing incorrectly
Why it's wrong: In , students may forget to distribute to both terms.
Correct: , not
Confusing parallel and perpendicular slopes
Why it's wrong: Parallel lines have the SAME slope. Perpendicular slopes are NEGATIVE RECIPROCALS.
Correct: Parallel: same . Perpendicular: if , then
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Practice Problems
18 problemsWhat is the point-slope form of a linear equation?
Why It Matters
- Finding parallel lines: Parallel lines have the same slope. Given any point, you can instantly write the equation of a parallel line.
- Finding perpendicular lines: If you know the original slope, the perpendicular slope is its negative reciprocal.
- Real-world modeling: When you have initial data (a point) and a rate of change (slope), point-slope form captures the relationship directly.
- Problem solving: Many word problems give you a starting value and a rate - perfect for point-slope form!
Real World Applications
Business Growth Projection
Companies use linear models to project growth when they know their current position and rate of change.
Example:
A startup has 500 users today and gains 50 new users per day. The equation models their user count, where is days from today.
A coffee shop made 200 euros in profit on day 3. They estimate their daily profit increases by 25 euros each day.
Write an equation to model their profit over time.
Step 1: Write the mathematical expression
Use point-slope form with the point (3, 200) and slope 25:
Temperature Change
Scientists model temperature changes using linear equations when the rate is constant.
Example:
If the temperature at 2 PM was 18°C and drops 3°C per hour, the equation models the temperature, where is the hour (in PM).
At 10 AM, a lab recorded a temperature of 25°C. The cooling system decreases temperature by 2°C per hour.
Write an equation for temperature after hours past 10 AM.
Step 1: Write the mathematical expression
Point: , slope:
Key Takeaways
- 1Point-slope form is where is slope and is a known point
- 2Use this form when you know a point and the slope of a line
- 3To convert to slope-intercept form, distribute and solve for
- 4For parallel lines, use the same slope with a new point
- 5For perpendicular lines, use the negative reciprocal of the slope
Frequently Asked Questions
Glossary
- Point-slope form
- A form of linear equation: , using a point and slope
- Slope
- The steepness of a line, calculated as rise over run:
- Parallel lines
- Lines that never intersect; they have the same slope
- Perpendicular lines
- Lines that intersect at a 90° angle; their slopes are negative reciprocals
- Negative reciprocal
- For slope , the negative reciprocal is . Example: negative reciprocal of is
Formula Card
Point-Slope Form
Where $m$ = slope, $(x_1, y_1)$ = known point
Slope Formula
Calculate slope from two points
Converting to Slope-Intercept
Distribute and solve for $y$ to convert