System of Equations Calculator
Solve Linear Systems Online
Free online system of equations calculator that solves 2x2 and 3x3 systems of linear equations using multiple methods. Get step-by-step solutions showing substitution, elimination, and Cramer's Rule methods with detailed steps and solution verification.
System of Equations Calculator
Our system of equations calculator helps you solve multiple equations simultaneously, whether you're working with 2x2 or 3x3 systems. Choose from three powerful solving methods: substitution, elimination (Gaussian elimination), or Cramer's Rule. Perfect for solving linear systems in algebra, business problems, or scientific calculations, this calculator shows every step of the solution process.
What is a System of Linear Equations?
A system of linear equations consists of two or more equations that must be solved together. Our calculator handles both 2x2 systems (two equations, two variables) and 3x3 systems (three equations, three variables). For example, in a 2x2 system:
2x + y = 10
x - y = 4
Our calculator can solve this using substitution, elimination, or Cramer's Rule, showing all steps.
Solving Methods Available
- Elimination Method (Gaussian Elimination): Also called addition method, this technique adds or subtracts equations to eliminate variables. For example, if one equation has 2x and another has -2x, adding them cancels out x completely. Works well when coefficients can be easily matched. Best for systems where variables can be easily eliminated by adding/subtracting equations.
- Substitution Method: Solve one equation for a variable (like x = 3 - 2y) and plug this expression into other equations. This replaces x with its equivalent expression, turning two variables into one. Most intuitive for beginners. Best for systems where one variable can be easily isolated, especially when one coefficient is 1.
- Cramer's Rule: Uses determinants (special number calculations from the coefficients) to find solutions. While more complex to understand, it's systematic and works reliably for most systems. Particularly useful for 3x3 systems where other methods become unwieldy. Best for larger systems (3x3) and when you need exact solutions rather than decimal approximations.
Types of Systems We Can Solve
- 2x2 systems of linear equations (two equations, two unknowns)
- 3x3 systems of linear equations (three equations, three unknowns)
- Systems with integer or decimal coefficients
- Word problems converted to systems of equations
- Systems solved by elimination or substitution
Features of Our Calculator
Unlike basic system of equations calculators, our tool provides comprehensive solutions using multiple methods. Whether you prefer solving systems of equations by elimination, substitution, or Cramer's Rule, you'll see every step clearly explained. The calculator handles both 2x2 and 3x3 systems, making it perfect for algebra, economics, and science applications.
How to Use
- Choose your system size (2x2 or 3x3 system of equations)
- Enter the coefficients and constants for each equation
- Select your preferred solving method (elimination, substitution, or Cramer's Rule)
- Click 'Solve' to see the complete step-by-step solution
- Review the detailed steps and verify your solution
Watch Out For
- Forgetting to include all variables when solving systems of linear equations
- Mixing up signs when using elimination or substitution methods
- Not checking if solutions make sense in the context of word problems
- Overlooking special cases like no solution or infinite solutions in linear systems
- Incorrect matrix operations when solving 3x3 systems
FAQs
What types of systems can this calculator solve?
Our calculator handles both 2x2 and 3x3 systems of linear equations, providing step-by-step solutions using Cramer's Rule.
Can it solve systems of equations with 3 variables?
Yes, our calculator can solve 3-variable systems of equations (3x3 systems) with detailed steps showing the solution process.
How accurate are the solutions?
Solutions are calculated to 4 decimal places for precision while maintaining readability. All steps are shown to help verify accuracy.
What if I've never solved systems of equations before?
Don't worry! Our calculator shows every step clearly. Start with the substitution method, which is usually easiest to follow. Each step is explained in plain language, and you can always check if your answer works by plugging it back into the original equations.