Equation of a Line with Two Points- Tutorial
What You're Getting Into
This tutorial shows you how to find the equation of a line when you know two points on that line. No theory lectures. Just the steps.
What you need: Two points with coordinates like (x₁, y₁) and (x₂, y₂). That's it.
The Slope-Intercept Form
Most people use y = mx + b to write line equations. Here's what each part means:
- m = slope (rise over run)
- b = y-intercept (where the line crosses the y-axis)
Your job is to find m first, then find b.
Step 1: Find the Slope
The slope formula is:
m = (y₂ - y₁) / (x₂ - x₁)
Take the difference in y-coordinates and divide by the difference in x-coordinates. Order matters, but stay consistent.
Example
Points: (2, 3) and (6, 11)
m = (11 - 3) / (6 - 2) = 8 / 4 = 2
Step 2: Find the Y-Intercept
Once you have the slope, plug it into y = mx + b. Then substitute one of your points to solve for b.
Using the slope (2) and point (2, 3):
3 = 2(2) + b
3 = 4 + b
Step 3: Write the Equation
Put it together: y = 2x - 1
Done. That's your answer.
The Point-Slope Form (Alternative)
Sometimes the point-slope form is faster, especially if you don't need to solve for the y-intercept:
y - y₁ = m(x - x₁)
Using point (2, 3) and slope 2:
y - 3 = 2(x - 2)
This is valid. You can leave it like this or expand it to slope-intercept form.
How To: Finding the Equation of a Line
- Identify your two points. Label them (x₁, y₁) and (x₂, y₂).
- Calculate the slope. Use m = (y₂ - y₁) / (x₂ - x₁).
- Pick one point and plug the slope into the point-slope equation: y - y₁ = m(x - x₁).
- Solve for y if you need slope-intercept form.
Common Mistakes
- Subtraction errors. Watch your signs when calculating slope. (11 - 3) is not the same as (3 - 11).
- Vertical lines. If x₁ = x₂, the slope formula breaks (division by zero). A vertical line through x = 5 is simply x = 5. No slope-intercept form exists.
- Horizontal lines. If y₁ = y₂, the slope is 0. The line is y = [that y-value].
Quick Comparison
| Method | Best When | Formula |
|---|---|---|
| Slope-Intercept | You need y = mx + b format | y = mx + b |
| Point-Slope | You have slope and one point | y - y₁ = m(x - x₁) |
| Two-Point Formula | Skip the middle step | (y - y₁) / (x - x₁) = (y₂ - y₁) / (x₂ - x₁) |
Worked Example
Problem: Find the equation of the line passing through (1, 5) and (4, 14).
Step 1: Slope
m = (14 - 5) / (4 - 1) = 9 / 3 = 3
Step 2: Y-intercept
5 = 3(1) + b
5 = 3 + b
Step 3: Equation
y = 3x + 2
Verify with the second point: 3(4) + 2 = 14 ✓