Lesson 7.7

Elimination (Multiplication)

What if nothing matches? What if they aren't even opposites? We can force a match by multiplying one equation by a "scaling factor."

Introduction

If you have and , they don't cancel. But if you multiply the by , it becomes . Now they are ready to fight!

Past Knowledge

Lesson 2.4 (Two-Step Equations). You need to be comfortable deciding what number to multiply by to reach a target.

Today's Goal

Scale one equation up to create additive inverses, then add.

Future Success

This is basically finding a "Common Denominator" but for equations.

Key Concepts

The "Scale Up" Strategy

Look for a relationship. Can turn into ? Yes, multiply by .

× -3

Target: Eliminate X

We want the top to cancel the bottom .

The Transformation

Notice: We multiplied EVERY term by -3.

Worked Examples

Example 1: Scaling X

Basic

Solve the system.

Step 1: Multiply Top by -2

We need to kill the positive . So we create a .

Step 2: Add to Bottom Eq

Step 3: Finish

Solution:

Example 2: Scaling Y

Intermediate

Solve the system.

Step 1: Multiply Bottom by -3

We want to cancel the . Multiply the bottom by .

Step 2: Add

Solution

Example 3: Choosing the "Easier" Path

Advanced

Which variable should we kill?

Option A: Eliminate X (Winner)

Identify (coefficient 1). Multiply by .

Add Top Eq:

Big numbers (), but easy decision!

Option B: Eliminate Y

Turn into . Multiply by .

Add Bottom Eq:

Also works! But requires more mental math.

Conclusion

Both methods give the valid solution . Option A is preferred because scaling by eliminates the variable immediately without worrying about the .

Common Pitfalls

Forgetting the C value

When you multiply the equation by 3, you must multiply the number after the equal sign too! must become . If you forget, the whole answer is wrong.

Choosing the Hard Way

Don't try to make and match yet. That requires multiplying BOTH equations (next lesson). Look for the easy match first.

Real-Life Applications

Scaling Recipes:

If you need 4 cups of flour for 1 cake, you need 12 cups for 3 cakes.
You "scale up" the inputs to match the desired output.
In Elimination, we "scale up" the equations to match the other variable so we can destroy it.