Simultaneous Equations Solver - Solve 2×2 & 3×3 Systems
Solve systems of 2 or 3 linear equations with multiple variables. Get step-by-step solutions using elimination, substitution, or matrix methods.
System of Equations
Solution
What is a Simultaneous Equations Solver?
A Simultaneous Equations Solver is a mathematical tool that finds the values of multiple variables that satisfy all given equations at the same time. It solves systems of linear equations with 2 or 3 unknowns.
This calculator works for:
- 2×2 Systems - Two equations with two variables (x and y)
- 3×3 Systems - Three equations with three variables (x, y, and z)
- Multiple Methods - Elimination, substitution, and matrix approaches
To solve quadratic equations, try our Quadratic Equation Solver with step-by-step solutions.
For completing the square method, check our Completing the Square Calculator to convert to vertex form.
To calculate slope and intercepts, use our Slope & Intercept Calculator for linear equations.
How the Solver Works
For 2×2 systems, the calculation uses Cramer's Rule:
y = (ae - cf) / (ad - bc)
Where:
- Determinant (D) = ad - bc
- D_x = ef - bd
- D_y = ae - cf
Key Concepts Explained
Determinant
A number calculated from coefficients that determines if a unique solution exists.
Elimination Method
Adding or subtracting equations to eliminate one variable at a time.
How to Use This Calculator
Choose System Size
Select 2×2 or 3×3 system
Enter Coefficients
Input all equation coefficients
Click Solve
Get solutions and steps
Benefits of Using This Calculator
- • Instant Solutions: Get answers for complex systems in seconds.
- • Step-by-Step Process: Understand the solution method with detailed steps.
- • Multiple Systems: Solve both 2×2 and 3×3 systems with one tool.
Factors That Affect Your Solution
1. Determinant Value
If determinant = 0, the system may have no solution or infinite solutions.
2. Equation Dependency
If equations are multiples of each other, solutions may be infinite.
Frequently Asked Questions (FAQ)
Q: What are simultaneous equations?
A: Simultaneous equations are a set of two or more equations with multiple variables that are solved together to find values that satisfy all equations at the same time.
Q: What methods can be used to solve simultaneous equations?
A: Common methods include elimination (adding or subtracting equations), substitution (solving for one variable and substituting), and matrix methods (using Cramer's rule or Gaussian elimination).
Q: Can simultaneous equations have no solution?
A: Yes, if the equations represent parallel lines (for 2×2) or parallel planes (for 3×3), they will have no solution. This occurs when the equations are inconsistent.
Q: What is Cramer's Rule?
A: Cramer's Rule is a method for solving systems of linear equations using determinants. It provides explicit formulas for each variable using ratios of determinants.
Q: When do simultaneous equations have infinite solutions?
A: Simultaneous equations have infinite solutions when all equations represent the same line or plane. This happens when the equations are dependent (one is a multiple of another).