Linear Combination Calculator
Solve systems of linear equations using the elimination method with step-by-step solutions
System of Linear Equations
First Equation
Format: a₁x + b₁y = c₁
Second Equation
Format: a₂x + b₂y = c₂
Solution
Enter coefficients for both equations to solve the system
Example: 2x + 3y = 7 and x - y = 1
Example: Linear Combination Method
System of Equations
x - y = 1
Solution Steps
Step 1: Multiply equation 2 by -2 to eliminate x
Step 2: 2x + 3y = 7 and -2x + 2y = -2
Step 3: Add equations: 5y = 5, so y = 1
Step 4: Substitute: x - 1 = 1, so x = 2
Solution: x = 2, y = 1
Linear Combination Method
Elimination
Multiply equations by constants to eliminate one variable
Create opposite coefficients for one variable
Addition
Add the modified equations together
One variable cancels out
Substitution
Solve for remaining variable, then substitute back
Find both variable values
Solution Types
Unique Solution
One point where lines intersect (x, y)
No Solution
Parallel lines (0 = c, where c ≠ 0)
Infinite Solutions
Same line (0 = 0 after elimination)
LCM Method
m₁ = LCM(a₁,a₂)/a₁, m₂ = -LCM(a₁,a₂)/a₂
Understanding Linear Combination Method
What is Linear Combination?
The linear combination method (also called elimination method) solves systems of linear equations by strategically combining equations to eliminate variables. This systematic approach reduces complex systems to simpler ones that can be solved step by step.
How It Works
- •Multiply: Scale equations by appropriate constants
- •Add: Combine equations to eliminate one variable
- •Solve: Find the remaining variable
- •Substitute: Back-solve for other variables
Applications
- •Engineering systems analysis
- •Economics and optimization
- •Physics and chemistry balancing
- •Computer graphics transformations
- •Network flow problems
Advantage: The linear combination method is often more efficient than substitution for systems where coefficients have common factors or are already opposites.
Strategy Selection
When to Use Linear Combination:
- • Coefficients are integers with common factors
- • Variables have opposite or equal coefficients
- • Fractions would complicate substitution
- • Multiple variables need elimination
Multiplier Selection:
- • Use LCM method for general cases
- • Use ±1 when coefficients are equal/opposite
- • Choose strategy that avoids fractions
- • Eliminate variable with simpler coefficients