Lesson 2.9

Variables on Both Sides (Simple)

You can't solve a puzzle if the pieces are in two different rooms. To solve for , we must first gather all the onto one side of the equation.

Introduction

In , the variable appears on both the left and right. This isn't allowed! We need to move one of them so they can be combined.

Past Knowledge

You know how to move constants (numbers) from one side to the other.

Today's Goal

Move variable terms across the equals sign using inverse operations.

Future Success

This skill is critical for finding the intersection of two lines (Unit 7).

Key Concepts

Move the "Little Guy"

To avoid negative numbers, always move the smaller variable term to the side of the larger variable term.

Move (smaller) to (larger).

Subtract

Move (smaller) to (larger).

Add

Wait... Constants or Variables First?

Technically, you can do either. But following this order makes it easier:

  1. Move Variables: Get all letters on one side first.
  2. Solve: You are left with a simple division or two-step equation.

Worked Examples

Example 1: Moving Right to Left

Basic

Solve for :

1

Move the Smaller Variable

Subtract from both sides to combine with .

==
Divide by 2:

Example 2: Moving Left to Right

Intermediate

Solve for :

1

Move the Smaller Variable

Subtract (smaller) to join (larger).

==
Divide by 5:

Example 3: Moving Negatives

Advanced

Solve for :

1

Move the Negative Variable

is smaller than . Add to both sides.

==
Divide by 10:

Common Pitfalls

Leaving Nothing Behind

If you move the only term on a side (e.g., ), remember you are left with . Don't leave the side blank!

Moving the Wrong Way

You strictly can't "combine" terms across the equals sign just by looking at them. You MUST perform the inverse operation (add/subtract) to move them.

Real-Life Applications

Comparing rental plans relies on this. If Plan A is $50/month and Plan B is $30/month plus a $100 setup fee:

To find the break-even point (), you have to subtract from both sides (), showing that after 5 months, they cost the same.

Practice Quiz

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