# Elimination Method — Steps and Worked Examples

## What Is The Elimination Method?  
The **elimination method** is a way to solve a system of equations by combining the equations so that one variable is removed. You add or subtract the equations to cancel a variable, solve the resulting one-variable equation, then substitute back to find the other variable. It is also called the **addition method**.

It works because adding equal things to equal things keeps an equation true: if a=ba = ba=b and c=dc = dc=d, then a+c=b+da + c = b + da+c=b+d. Cancelling a variable is just a clever choice of what to add.

### When do you add, and when do you subtract?  
The decision rests on the coefficients of the variable you want to remove.
- **Add** the equations when the matching coefficients are **opposites** (like +3y and −3y). They sum to zero.
- **Subtract** one equation from the other when the matching coefficients are **equal** (like 2x and 2x). They cancel on subtraction.

If neither matches, you first multiply one or both equations by a constant to _make_ a pair match. That scaling step is the heart of the method.

## The Steps Of The Elimination Method
| Step | What you do |
| --- | --- |
| 1 | Write both equations in standard form ax+by=c, lined up by variable. |
| 2 | Multiply one or both equations so one variable has matching or opposite coefficients. |
| 3 | Add (opposite coefficients) or subtract (equal coefficients) to eliminate that variable. |
| 4 | Solve the resulting single-variable equation. |
| 5 | Substitute that value back into either original equation to find the other variable. |
| 6 | Check the pair in both original equations. |

## Examples Of The Elimination Method

### Example 1  
**Solve:**  
x+y=7  
x−y=3  
The y coefficients are opposites (+1 and −1), so add the equations:

2x=10  
x=5  
Substitute into the first equation:  
5+y=7  
y=2  
Final answer: (5,2).

### Example 2  
**Solve:**  
3x+2y=16  
3x−y=4  
The x coefficients are equal (3 and 3). The first instinct is to _add_ the equations — but watch what adding does:  
6x+y=20  
Both variables survived; nothing was eliminated. The fix is to subtract:

(3x+2y)−(3x−y)=16−4  
3y=12  
y=4  
Substitute into the second equation:  
3x−4=4  
x=8  
Final answer: (8, 4).

### Example 3  
**Solve (coefficients must be scaled first):**  
2x+3y=13  
4x−y=5  
Multiply the second equation by 3 so the y terms become opposites:

2x+3y=13  
6x−3y=15  
Now add:

8x=28  
x=2  
Substitute into the second original equation:

4(2)−y=5  
y=3  
Final answer: (2,3).

### Example 4  
**Solve (scale both equations):**  
3x+4y=10  
2x+5y=9  
To cancel x, multiply the first by 2 and the second by 3:

6x+8y=20  
6x+15y=27  
Subtract:  
7y=7  
y=1  
Substitute into 3x+4y=10:

3x+4=10  
x=2  
Final answer: (2,1).

### Example 5  
**A no-solution case:**  
2x+3y=12  
4x+6y=30  
Multiply the first by 2 to match the x coefficients:

4x+6y=24  
Subtract:
0=−6  
Final answer: **no solution** (inconsistent system).

### Example 6  
**Two numbers have a sum of 9 and a difference of 5. Find them.**
Let the numbers be x and y:

x+y=9  
x−y=5  
Add to cancel y:

2x=14  
x=7  
y=2  
Final answer: the numbers are 7 and 2.

## Why The Elimination Method Earns Its Place  
The reason to reach for elimination is that it stays clean where other methods get messy.
- **No fractions until the end.** When both equations are in standard form ax+by=c, elimination avoids the awkward fractions that substitution often introduces mid-solve.
- **It is the standard form for machines.** Gaussian elimination — the same add-and-cancel idea applied row by row — is how computers solve large systems.

## Where Students Lose The Mark On Elimination
### Mistake 1: Adding when you should subtract (or vice versa)  
**The correct way:** Equal coefficients cancel on _subtraction_; opposite coefficients cancel on _addition_.
### Mistake 2: Sign errors when subtracting  
**The correct way:** Distribute the minus across _every_ term: rewrite the subtracted equation with all signs flipped first, then add.
### Mistake 3: Forgetting to back-substitute, or substituting into the scaled equation  
**The correct way:** After solving for one variable, substitute back into one of the _original_ equations to find the other.

## Conclusion  
- The **elimination method** solves a system by adding or subtracting equations to cancel one variable.
- A false statement after both variables cancel means no solution; a true statement means infinitely many.

## Practice Questions on the Elimination Method  
1. By adding: x+y=10 and x−y=4.  
2. By subtracting: 2x+y=7 and 2x+3y=11.  
3. By scaling first: 3x+2y=7 and 5x+4y=13.  
4. Classify: 2x+3y=12 and 4x+6y=18.

**Answers**  
1. Final answer: (7,3).  
2. Final answer: (2,2).  
3. Final answer: (1,2).  
4. Final answer: **no solution**.