Lesson 6.6

Writing Equations (The Fast Way)

Why solve for if you don't have to? Point-Slope form lets you write the equation in one single step.

Introduction

In Chapter 14, we spent a lot of time finding . It was slow. Today, we speed it up. If you have a slope and a point, you can write the equation instantly.

Past Knowledge

Lesson 6.5 (Point-Slope Form). You need to know .

Today's Goal

Write the equation of a line without doing any algebra calculations.

Future Success

This is the preferred method in higher-level math like Calculus because it's faster.

Key Concepts

One-Step Writing

If you are given:

Slope:
Point:

You just construct the skeleton:

That's it. You are done.

Worked Examples

Example 1: Basic Writing

Basic

Write the equation of the line with slope passing through .

Step 1: Plug in

on the left.

on the right.

Example 2: Negative Coordinates

Intermediate

Write the equation of the line with slope passing through .

Step 1: Plug in

on the left.

on the right.

Step 2: Clean up Signs

Example 3: Writing from Two Points

Advanced

Write equation for line through and .

Step 1: Find Slope

Step 2: Pick Point & Write

Using :

Both answers are correct!

Common Pitfalls

Over-working

If the question asks for "Point-Slope Form", DO NOT DISTRIBUTE. If you change it to , you might lose points for not following instructions (and you did extra work!).

Thinking they are different

In Example 3, the two equations LOOK different, but if you simplify both, they become the exact same line. They are synonyms.

Real-Life Applications

Computer Graphics:

Lines are defined by vertices (points).
Algorithms often use the point-slope form internally to calculate pixel positions because finding the y-intercept (which might be way off-screen) is inefficient.