Deriving the Equation of a Line

Lesson 2 of 2

In this lesson:

• Derive for a line through the origin
• Derive for a line with y-intercept
• Connect and to their geometric meanings

Learning Objectives for This Deck

You should be able to:

1. Derive for a line through the origin
2. Derive for a line with y-intercept
3. Connect to slope and to the y-intercept

Use the Easiest Slope Triangle

Slope is constant — so pick the simplest triangle.

• One corner at the origin
• The other at a general point

What equation must every point on the line obey?

From the Origin to a General Point

Why the Rise Is y From the Origin

Starting at :

• The vertical leg climbs from height up to height
• So the rise is , and the run is

Test the Equation Against Points

Does describe the line?

• : ✓
• : ✓

Every point on the line fits.

Generalize to Any Origin Line

For any line through the origin with slope :

• The Grade-7 constant is the slope

A Negative-Slope Example Through Origin

Line through the origin with slope :

• ; test : ✓

Solo Check: Write the Equation

A line through the origin passes through .

Write its equation on your own paper.

When the Origin Equation Fails

A line has slope but crosses the y-axis at .

• At , predicts
• But the actual point is — off by 3

The Subtlety: Rise Is y − b

• From the rise is , not

Derive the Equation With a Y-Intercept

From to with slope :

• Test : ✓ · : ✓

Generalize to Any Slanted Line

For a line with slope and y-intercept :

The Origin Line as a Special Case

When :

• Lines through the origin are included — they just start at height

A Line With a Negative Y-Intercept

Slope , y-intercept :

• Rise , so

What m and b Mean

In :

• is the slope — steepness and direction
• is the y-intercept — the height where

Set : . The point is on the y-axis.

Your Turn: Graph to Equation

Here is a line crossing the y-axis at with slope .

Identify , build a slope triangle for , write , and verify a point.

Unscaffolded: Build the Equation Yourself

Here is a new line with a non-zero y-intercept.

Find , find with a slope triangle, write the equation, and verify two points — no steps given.

Watch Out: Two Common Errors

Rise is , not — from you start at height , so subtract it
is the y-intercept, not the x-intercept — set to get

Key Takeaways From This Lesson

✓ A line's equation is its slope triangle solved for
✓ Origin: · Y-intercept :
✓ = steepness, = starting height; is the case

Coming Up Next in This Unit

You can now derive any non-vertical line's equation. Next you'll put two of these together to find where lines cross — systems of equations — and treat as a linear function.