Lesson 6.12

Writing Parallel Equations

Now that we know parallel lines have the same slope, let's practice writing their equations from scratch.

Introduction

We are going to give you a "reference line" and a point. Your job is to ignore the reference line's position but steal its direction.

Past Knowledge

Lesson 6.6 (Writing Point-Slope). You need to be fast at writing equations.

Today's Goal

Master the 3-step process: Find Slope, Keep Slope, Write Equation.

Future Success

In calculus, you'll find tangent lines parallel to secant lines (Mean Value Theorem).

Key Concepts

The "Copy-Paste" Method

  1. Identify Old Slope ().

    If the equation is messy (Standard Form), solve for first.

  2. Keep It ().

    Do not change the sign. Do not flip it. Keep it exactly the same.

  3. Use New Point.

    Plug the slope and the NEW point into .

Worked Examples

Example 1: The Standard Problem

Basic

Write equation parallel to going through .

Step 1: Steal Slope

Old line has .

Step 2: Write Equation

Use and .

Example 2: Hidden Slope

Intermediate

Write equation parallel to going through .

Step 1: Find Slope

Rewrite: .

Slope is (coefficient of x).

Step 2: Write Equation

Use and .

Example 3: Special Cases

Advanced

Write equation parallel to going through .

Step 1: Identify Type

is a vertical line. It has undefined slope.

Step 2: Copy Format

A line parallel to a vertical line is also vertical.

Use the x-coordinate of the new point ().

Common Pitfalls

Stealing "b"

Do not accidentally use the old y-intercept. The new line is in a completely different location. The ONLY thing they share is the slope.

Negating Slope

Some students mix up parallel and perpendicular rules. For parallel, do NOTHING to the slope. Just copy it.

Real-Life Applications

Graphic Design:

  • When creating an "offset" border or a shadow effect for text, you are essentially drawing a parallel figure slightly shifted.
  • The vector math software calculates a new path that maintains a constant distance from the original path.

Practice Quiz

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