Lesson 7.8

Elimination (Complex)

The final boss of systems. What if nothing matches, and you can't easily turn one number into another? We have to change BOTH equations.

Introduction

If you have and , you can't turn 2 into 3 easily. Instead, we find a "Least Common Multiple" (LCM) like 6. We multiply the top by 3 and the bottom by 2.

Past Knowledge

Lesson 1.12 (Fraction Operations). Finding a Common Denominator is exactly the same skill.

Today's Goal

Multiply TWO equations to create an opposite pair.

Future Success

This works for ANY system. It is the universal hammer. If you master this, you can solve anything.

Key Concepts

The "Criss-Cross" Multiplication

To eliminate and , simply multiply the top equation by and the bottom equation by .

× 5

× 2

Multiply top by the bottom number (). Multiply bottom by the top number ().

The Result (Matches!)

See? We created and . Perfect opposites.

Worked Examples

Example 1: Opposite Signs (Easy)

Basic

Solve the system.

Step 1: Target Y

Top multiply by 3. Bottom multiply by 4.

Step 2: Add

Step 3: Solution

Plug into top eq: .

→ → .

Solution:

Example 2: Same Signs (Need Negative)

Intermediate

Solve the system.

Step 1: Choose Y

Top multiply by . Bottom multiply by (to flip sign).

Step 2: Add

Solution

→ → .

Solution:

Example 3: Prime Numbers

Advanced

Sometimes you just get big numbers. Don't panic.

Option A: Target Y (Winner)

LCM of and is . Small numbers!

Option B: Target X

LCM of and is . Getting big...

Conclusion

Both options work perfectly. However, Option A keeps the numbers smaller () compared to Option B (). Smaller numbers = fewer mistakes!

Common Pitfalls

Forgetting the Negative

If the signs are already the same, you MUST multiply one equation by a NEGATIVE number. If you don't, you'll end up adding (), which kills nothing.

Messy Handwriting

This involves rewriting two equations entirely. Write clearly. Line up your columns. If you are messy, you will fail this lesson.

Real-Life Applications

Crypto-Currency & Security:

Encryption algorithms rely on finding large prime factors and solving complex systems.
While computers use matrices (which are just giant elimination machines), the logic of scaling rows to cancel out values is the foundation of linear algebra.