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.

Intermediate25 minLesson
Printable worksheet

Definition

The point-slope form of a linear equation is:
y−y1=m(x−x1)y - y_1 = m(x - x_1)
Where:
  • mm is the slope of the line
  • (x1,y1)(x_1, y_1) is a specific point on the line
  • (x,y)(x, y) represents any other point on the line
This form is especially useful when you know:
  1. 1.The slope of a line, AND
  2. 2.One point that the line passes through

Try it now

What is the point-slope form of a linear equation?

Worked Examples

Write the equation of a line with slope m=3m = 3 that passes through the point (2,5)(2, 5).
1

Identify the values

Slope m=3m = 3, Point (x1,y1)=(2,5)(x_1, y_1) = (2, 5) → m=3m = 3, x1=2x_1 = 2, y1=5y_1 = 5
2

Write point-slope form

y−y1=m(x−x1)y - y_1 = m(x - x_1) → Template ready
3

Substitute values

y−5=3(x−2)y - 5 = 3(x - 2)

Common Mistakes

Forgetting to change the sign when x1x_1 or y1y_1 is negative

Why it's wrong: In y−y1=m(x−x1)y - y_1 = m(x - x_1), if y1=−3y_1 = -3, then y−(−3)=y+3y - (-3) = y + 3, not y−3y - 3.

Correct: Always write the subtraction first, then simplify: y−(−3)=y+3y - (-3) = y + 3

Using the wrong point for (x1,y1)(x_1, y_1)

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 y−5=3(x−2)y - 5 = 3(x - 2), students may forget to distribute to both terms.

Correct: 3(x−2)=3x−63(x - 2) = 3x - 6, not 3x−23x - 2

Confusing parallel and perpendicular slopes

Why it's wrong: Parallel lines have the SAME slope. Perpendicular slopes are NEGATIVE RECIPROCALS.

Correct: Parallel: same mm. Perpendicular: if m=23m = \frac{2}{3}, then m⊥=−32m_{\perp} = -\frac{3}{2}

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Point-Slope Form | Equation of a Line with Examples

by Understand The Math

Watch on YouTube

Interactive Visual

Function graph

-4-202468-50510xy(2, 5)

Tap or hover over the graph to read off values.

y - 5 = m(x - 2)

Linear Function Explorer

y = 3x - 1
Slope (m)3
Y-Intercept (b)-1
b
run
rise

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
What is the point-slope form of a linear equation?

Why It Matters

Point-slope form is a powerful tool for writing equations quickly:
  • 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 y−500=50(x−0)y - 500 = 50(x - 0) models their user count, where xx is days from today.

1Try It Yourself

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 T−18=−3(t−2)T - 18 = -3(t - 2) models the temperature, where tt is the number of hours after noon.

2Try It Yourself

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 TT after hh hours past 10 AM.

Step 1: Write the mathematical expression

Point: (0,25)(0, 25), slope: −2-2

Key Takeaways

  • 1Point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1) where mm is slope and (x1,y1)(x_1, y_1) 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 yy
  • 4For parallel lines, use the same slope with a new point
  • 5For perpendicular lines, use the negative reciprocal of the slope

Frequently Asked Questions

Use point-slope form when you know a point and the slope but don't know the y-intercept. It's quicker than finding bb first.
Use point-slope form when you know a point and the slope but don't know the y-intercept. It's quicker than finding bb first.
No! Using either point will give you an equivalent equation. The final line is the same - just the form looks different until you simplify.
Two lines are parallel if and only if they have the same slope but different y-intercepts. In point-slope form, the mm values must be equal.

Glossary

Point-slope form
A form of linear equation: y−y1=m(x−x1)y - y_1 = m(x - x_1), using a point and slope
Slope
The steepness of a line, calculated as rise over run: m=ΔyΔxm = \frac{\Delta y}{\Delta x}
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 mm, the negative reciprocal is −1m-\frac{1}{m}. Example: negative reciprocal of 23\frac{2}{3} is −32-\frac{3}{2}

Formula Card

Point-Slope Form

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Where $m$ = slope, $(x_1, y_1)$ = known point

Slope Formula

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Calculate slope from two points

Converting to Slope-Intercept

y=mx+by = mx + b

Distribute and solve for $y$ to convert

Today's challengeOne problem, three tries

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