Elimination Method Calculator
Free elimination method calculator for solving systems of linear equations. Get step-by-step solutions with the addition/subtraction method and variable elimination. Perfect for algebra students learning to solve systems using elimination.
Last updated: December 15, 2024
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Enter in the form: ax + by = c
Enter in the form: ax + by = c
Solution Found
x-value
x = 2
y-value
y = 3
Eliminated Variable:
y
Step-by-Step Solution:
Verification:
Check: (2) + (3) = 5 ✓ and 2(2) - (3) = 1 ✓
Elimination Method Tips:
- • Add or subtract equations to eliminate one variable
- • Multiply equations by constants to make coefficients match
- • Look for variables with same or opposite coefficients
- • Always verify your solution in both original equations
- • If 0 = 0, infinite solutions; if 0 = non-zero, no solution
Elimination Method Applications & Types
Example system
x + y = 5
2x - y = 1
Add equations when coefficients are opposite
Example system
2x + y = 7
2x - y = 3
Subtract equations when coefficients match
Example system
2x + 3y = 8
3x - 2y = 1
Multiply to create matching coefficients
Result
0 = 5 (False)
Elimination leads to contradiction: no solution
Result
0 = 0 (True)
Elimination leads to identity: infinite solutions
Process
Multiple eliminations
Eliminate variables systematically, one at a time
Quick Example Result
For system: x + y = 5, 2x - y = 1
x-value
x = 2
y-value
y = 3
How the Elimination Method Works
The elimination method is a systematic algebraic approach to solving systems of equations by adding or subtracting equations to eliminate one variable. This method is particularly effective when variables have convenient coefficients that are equal or opposite, making it often faster than substitution for many systems.
The Elimination Process
This systematic approach ensures accuracy and helps identify special cases.
When to Use Elimination Method
The elimination method is most efficient when variables have coefficients that are equal, opposite, or easily made to match through multiplication. It's particularly useful when you want to avoid fractions during the solving process, as it often keeps calculations cleaner than the substitution method.
- Variables have the same coefficient (subtract equations)
- Variables have opposite coefficients (add equations)
- Coefficients are easily matched by simple multiplication
- System has integer coefficients that work well together
- You want to avoid the fractions common in substitution
Sources & References
- Elementary and Intermediate Algebra - Bittinger, Ellenbogen, Johnson (6th Edition)Comprehensive coverage of elimination method techniques
- College Algebra - Blitzer (7th Edition)Systems of equations and elimination method applications
- Khan Academy - Systems of Equations: EliminationStep-by-step video tutorials and practice problems
Need help with other algebra topics? Check out our substitution method calculator and quadratic formula calculator.
Get Custom Calculator for Your PlatformElimination Method Example
Given System:
Solution Steps:
- Step 1: Write the system of equations
- x + y = 5 ... (1)
- 2x - y = 1 ... (2)
- Step 2: Add equations to eliminate y
- (1) + (2): x + y + 2x - y = 5 + 1
- 3x = 6
- x = 2
- Step 3: Substitute x = 2 into equation (1)
- 2 + y = 5
- y = 3
Solution: x = 2, y = 3
Check: (2) + (3) = 5 ✓ and 2(2) - (3) = 1 ✓
Subtraction Example
3x + y = 10, 3x - y = 2
Subtract: 2y = 8 → y = 4, x = 2
Multiplication Example
2x + 3y = 7, 3x - y = 5
Multiply second by 3, then add
Frequently Asked Questions
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