Lesson 7.6

Elimination (Subtraction)

Multiplication gave us opposites. But what if the terms are identical twins? We need to subtract everything to make them cancel.

Introduction

If you have and , adding them gives . That doesn't help! Equal terms don't cancel. We need to Subtract the entire second equation.

Past Knowledge

Lesson 1.11 (Distributing Negatives). You are essentially multiplying the second equation by -1.

Today's Goal

Solve systems with identical terms by flipping signs and adding.

Future Success

Always adding is safer than subtracting. We prefer to "Flip and Add" to avoid sign errors.

Key Concepts

The "Flip and Add" Strategy

Instead of trying to subtract in your head (which causes errors), rewrite the second equation by multiplying EVERYTHING by -1. Then just add like before.

Original:

FLIP THE BOTTOM ROWS SIGNS

Worked Examples

Example 1: Identical Xs

Basic

Solve the system.

Step 1: Flip & Add

Multiply bottom by -1:

Add to top:

Step 2: Plug Back In

Solution:

Example 2: Negative Signs Tricky

Intermediate

Solve the system.

Step 1: Flip Signs

Bottom becomes:

Note: -2y became +2y!

Step 2: Add

cancels.

Solution

Plug in to find .

Example 3: Double Negatives

Advanced

Solve.

Step 1: Flip

Multiply bottom by -1:

Step 2: Add

. (Gone!)

Common Pitfalls

Partial Flipping

When you multiply the bottom row by -1, you MUST flip signs on the right side of the equals sign too! becomes .

Adding Identical Terms

If you forget to flip signs and just add , you get . You didn't eliminate anything! If your variable is still there, you did it wrong.

Real-Life Applications

Accounting Reconciliation:

Accountants look for discrepancies by comparing two spreadsheets.
They "subtract" this month's sheet from last month's.
The identical rows cancel out to zero. The only things remaining are the changes. That's elimination!