Lesson 2.10

Variables on Both Sides (Complex)

This is the "Boss Fight" of solving linear equations. We combine Distribution, Combining Like Terms, and Moving Variables into a single, powerful algorithm.

Introduction

When an equation has parentheses, messy terms, and variables everywhere, don't panic. Just follow the roadmap. We solve these layer by layer, starting from the inside out.

Past Knowledge

You have mastered Distribution, Like Terms, and Inverse Operations separately.

Today's Goal

Combine all skills to solve complex multi-step equations.

Future Success

This exact process is used in geometry, physics, and engineering calculations.

Key Concepts

The "D.C.M.M.D" Checklist

Memorize this order to solve ANY linear equation:

  1. Distribute (Parentheses)
  2. Combine Like Terms (Same Side)
  3. Move Variables (To One Side)
  4. Move Constants (Add/Sub)
  5. Divide (Multiply/Div)

"Don't Call Me Mr. Dumb"

Why this order?

You can't move terms trapped inside parentheses (Step 1).
It's messy to move pieces before piling them up (Step 2).
Once simplified, it's just a regular two-step equation (Steps 3-5).

Worked Examples

Example 1: Distribute & Move

Basic

Solve for :

1

Distribute

2

Move Variables

Subtract from both sides.

==
3

Solve Two-Step

Add 8, then divide by 2.

Example 2: Combine Before Moving

Intermediate

Solve for :

1

Distribute

2

Combine Like Terms

Combine and .

3

Solve

==
Divide by 5:

Example 3: Double Distribution

Advanced

Solve for :

1

Distribute Both Sides

2

Move Variables

Subtract from both sides.

3

Solve

==
Divide by 3:

Common Pitfalls

Skipping "Combine"

Students often try to move terms before combining them (e.g., subtracting from a side that has ). It works, but it causes errors. Simplify first!

Sign Flipping Errors

When moving a term like , remember to add it using . Don't subtract it again!

Real-Life Applications

Engineers solve forces on bridges using this. If the load on the left support is and the right is :

They distribute the 500, combine terms, and move variables to find the exact balancing point .

Practice Quiz

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