Elimination Method — Steps and Worked Examples
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
- By adding: x+y=10 and x−y=4.
- By subtracting: 2x+y=7 and 2x+3y=11.
- By scaling first: 3x+2y=7 and 5x+4y=13.
- Classify: 2x+3y=12 and 4x+6y=18.
Answers
- Final answer: (7,3).
- Final answer: (2,2).
- Final answer: (1,2).
- Final answer: no solution.