Lesson 7.5

Elimination (Addition)

Substitution is great, but sometimes it's faster to just destroy a variable.

Introduction

If you have and I give you , they cancel out to zero. We can use this property to eliminate a variable by simply adding two equations together vertically.

Past Knowledge

Lesson 2.7 (Combining Like Terms). You need to be comfortable adding vertical columns of terms.

Today's Goal

Solve systems by adding equations to cancel out a variable.

Future Success

This is the most powerful method for "standard form" equations ().

Key Concepts

The Vertical Drop

Look for Opposites. If you see and , they are ready to be eliminated.

0

Add down. The s disappear. Solve for . Easy!

Worked Examples

Example 1: Eliminating Y

Basic

Solve the system.

Step 1: Add Down

(CANCELLED)

Result:

Step 2: Solve

Step 3: Plug Back In

Use .

→ .

Solution:

Example 2: Eliminating X

Intermediate

Solve the system.

Step 1: Add Down

and cancel.

Result:

Step 2: Solve & Plug In

Plug into first equation:

→ →

Solution:

Example 3: Rearranging First

Advanced

Solve the system.

Step 1: Align

They don't line up! Move in the second equation.

. Wait... and don't cancel.

Actually, this one needs "Subtraction" which is next lesson. But let's verify we CANNOT add yet.

If we add: . Nothing cancelled. Stop.

Lesson Learned

You can ONLY use this method if the signs are already opposite (+ and -). If they are the same, don't add.

Common Pitfalls

Adding Unlike Terms

Make sure is under , and is under . If the equation is , simplify rearrange it to Standard Form () before adding.

Forgetting the Right Side

Don't forget to add the numbers after the equals sign too! . Students often forget this part.

Real-Life Applications

Chemistry Mixtures:

  • Balancing chemical equations is exactly this process.
  • You add molecular components to ensure mass is conserved.
  • Eliminating variables is how scientists figure out unknown reaction quantities.

Practice Quiz

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