Elimination Method Calculator
Solve systems of linear equations using the elimination method with step-by-step solutions
System of Linear Equations
Current System:
First Equation: a₁x + b₁y = c₁
Second Equation: a₂x + b₂y = c₂
Solution
Solution Type
Solution: x = 4.000000, y = 1.500000
Step-by-Step Solution
Example Problems
Example 1: Simple Elimination
System: 3x - 4y = 6, -x + 4y = 2
Method: Add equations directly (y coefficients are opposites)
Result: 2x = 8, so x = 4, y = 1.5
Example 2: Using Multipliers
System: 2x + 3y = 5, 2x + 7y = -3
Method: Multiply first equation by -1, then add
Result: 4y = -8, so y = -2, x = 5.5
Example 3: No Solution
System: -4x + 8y = 5, 3x - 6y = -1
Method: After elimination, get 0 = 11 (false)
Result: No solution (inconsistent system)
Elimination Method Steps
Arrange Equations
Ensure variables are aligned
Create Opposites
Multiply equations to get opposite coefficients
Add Equations
Eliminate one variable
Solve & Substitute
Find remaining variable, then substitute back
Solution Types
Unique Solution
One specific (x, y) pair
Infinite Solutions
Dependent equations (same line)
No Solution
Inconsistent system (parallel lines)
Understanding the Elimination Method
What is the Elimination Method?
The elimination method is a systematic approach to solving systems of linear equations by eliminating one variable at a time. By strategically adding equations together, we can eliminate a variable and reduce the system to a simpler form that's easier to solve.
When to Use Elimination
- •When coefficients are already opposites or can be made opposites easily
- •For systems with integer coefficients
- •When substitution would result in fractions or complex expressions
Key Strategies
Creating Opposites
Use LCM to find multipliers that create opposite coefficients
Choose Wisely
Eliminate the variable that requires simpler multipliers
Verify Solution
Always check your answer in both original equations
Special Cases
Unique Solution
Lines intersect at exactly one point
Infinite Solutions
Same line (dependent equations)
No Solution
Parallel lines (inconsistent system)