Elimination Calculator
Solve systems of linear equations using the elimination method. Get step-by-step solutions with detailed explanations.
Enter Equations in the form: ax + by + cz + ... = d
Solution
System Type
Verification
Step-by-Step Solution
Enter equations and click "Solve System" to see the step-by-step solution.
Elimination Method Steps
- 1. Write equations in standard form (ax + by + cz = d)
- 2. Multiply equations to match coefficients
- 3. Add or subtract to eliminate a variable
- 4. Repeat until one variable remains
- 5. Back-substitute to find other variables
Types of Solutions
- Unique Solution: One set of values satisfies all equations
- No Solution: Equations are inconsistent (parallel lines)
- Infinite Solutions: Equations are dependent (same line/plane)
Common Applications
- Engineering circuit analysis
- Economic equilibrium models
- Chemical reaction balancing
- Physics force analysis
- Business break-even analysis
Frequently Asked Questions
What is the elimination method?
The elimination method (also called the addition method) solves systems of linear equations by adding or subtracting equations to eliminate one variable at a time. This reduces the system to a single equation with one variable, which can then be solved directly.
How does Gaussian elimination work?
Gaussian elimination transforms the augmented matrix into row-echelon form using row operations: swapping rows, multiplying rows by constants, and adding multiples of one row to another. Back-substitution then finds the solution.
What's the difference between elimination and substitution?
Elimination adds/subtracts equations to remove variables, while substitution solves for one variable and plugs it into other equations. Elimination is often more efficient for larger systems.
Can elimination solve any system of linear equations?
Yes, elimination can solve any system of linear equations. It will produce either a unique solution, indicate no solution (inconsistent system), or show infinite solutions (dependent system).
What is the difference between Gaussian and Gauss-Jordan elimination?
Gaussian elimination stops at row-echelon form (upper triangular) requiring back-substitution. Gauss-Jordan continues to reduced row-echelon form (diagonal matrix), giving solutions directly without back-substitution.