The Elimination Method: Canceling Out the Chaos

When solving systems of equations, "Substitution" is like being a spy—quietly replacing one value with another. The Elimination Method (sometimes called the Addition Method) is more like a bulldozer. You stack the equations on top of each other and crush them together so that one variable completely disappears.

This method relies on a simple mathematical truth: If A = B and C = D, then A + C = B + D. You can add two true equations together to create a new true equation.

1. The Goal: Creating Opposites

To make a variable disappear (eliminate), you need Opposites.
What happens when you add 5x and -5x? You get 0. The x vanishes.

[Image of elimination method algebra steps]

Your goal is to manipulate the equations so that the x-terms (or y-terms) are exact opposites of each other.

2. Scenario A: The Ready-Made System

Sometimes, the equations are already perfectly set up for you.

System:
2x + y = 10
3x - y = 5

Step 1: Add Vertically

Notice the y terms (+y and -y) canceled out to zero!

Step 2: Solve

5x = 15
x = 3

Step 3: Plug Back In

Use any equation to find y.
2(3) + y = 10
6 + y = 10
y = 4
Solution: (3, 4)

3. Scenario B: Multiply One Equation

Sometimes the variables match in number but not in sign, or they don't match at all.

System:
x + 3y = 6
2x - 7y = -1

If we add these now, nothing cancels (3x - 4y... useless). We need opposites. Let's eliminate x. The bottom has 2x, so we want the top to be -2x.

Step 1: Multiply

Multiply the ENTIRE top equation by -2.
-2(x + 3y) = -2(6)
-2x - 6y = -12

Step 2: Add

Step 3: Solve

-13y = -13
y = 1

Plug y=1 back into the original top equation:
x + 3(1) = 6
x = 3
Solution: (3, 1)

4. Scenario C: Multiply Both Equations

This is the hardest type. Imagine trying to eliminate variables when you have 3x and 4x. You can't turn 3 into 4 easily. Instead, you find the Least Common Multiple (just like finding a common denominator).

System:
3x + 4y = 10
4x - 3y = 5

Let's eliminate y. We have +4y and -3y. The common multiple is 12.
- Multiply the top by 3 (to get +12y).
- Multiply the bottom by 4 (to get -12y).

Step 1: Transformation

Top: 9x + 12y = 30
Bottom: 16x - 12y = 20

Step 2: Add

25x = 50
x = 2

5. Substitution vs. Elimination: Which is Better?

While both methods always work, mathematicians are lazy. We pick the easiest one.

• Use Substitution when a variable is already isolated (y = ...).
• Use Elimination when both equations are in Standard Form (Ax + By = C). It saves you from dealing with messy fractions.

6. Conclusion

The Elimination Method is the heavy lifter of linear algebra. By keeping your equations aligned in neat columns and manipulating them with multiplication, you can make complex systems collapse into simple answers. The key is to always look for the "Opposite"—the number that will make the other disappear.