System of Equations Calculator
Solve systems of linear equations using multiple methods with step-by-step solutions
Setup System of Equations
Enter Coefficients
First Equation: a₁x + b₁y = d₁
Second Equation: a₂x + b₂y = d₂
Example Problems
Example 1: Simple 2x2 System
System:
3x - y = 0
2y - z = 25
Solution: x = 6, y = 18
Example 2: Cookie Riddle
Problem: 3 donuts = 1 cookie, 2 cookies - 1 candy = 25, 2 candies - 1 donut = 16
Variables: x = donut value, y = cookie value, z = candy value
System (rewritten):
3x - y = 0
2y - z = 25
-x + 2z = 16
Solution Methods
Gaussian Elimination
Row operations to create triangular form
Substitution
Solve for one variable, substitute into other equation
Elimination
Add/subtract equations to eliminate variables
Cramer's Rule
Use determinants to find solutions directly
System Tips
Enter coefficients carefully - order matters!
Use 0 for missing variables in equations
Determinant = 0 means no unique solution
Always verify solutions by substitution
Understanding Systems of Linear Equations
What is a System of Linear Equations?
A system of linear equations is a collection of two or more linear equations involving the same set of variables. We're looking for values of the variables that satisfy all equations simultaneously.
Types of Solutions
- •Unique Solution: Exactly one solution exists
- •No Solution: Equations are inconsistent
- •Infinite Solutions: Equations are dependent
Method Comparison
Gaussian Elimination
Most systematic, works for any size system
Substitution
Intuitive, good for small systems
Cramer's Rule
Direct formula using determinants