System of Equations Solver
Solve 2 × 2 and 3 × 3 systems of linear equations online. Find a unique solution or determine whether the system has no solution or infinitely many solutions.
This System of Equations Solver finds the values of x, y, and optionally z that satisfy all equations in a linear system simultaneously. It supports systems of 2 equations with 2 variables and 3 equations with 3 variables.
How to Use the System of Equations Solver
Choose the system size and enter the coefficient of every variable followed by the constant on the right side of each equation. Enter 0 when a variable does not appear in an equation, then select Solve System.
Entering a Linear Equation
Each row represents an equation in standard form. For example, 2x - 3y = 7 should be entered with 2 as the coefficient of x, -3 as the coefficient of y, and 7 as the constant.
What Is a System of Linear Equations?
A system of linear equations is a collection of equations containing the same variables. A solution is valid only when the variable values satisfy every equation in the system at the same time.
Unique Solution
A system has a unique solution when exactly one set of values satisfies every equation. For a two-variable system, this corresponds geometrically to two lines intersecting at one point.
No Solution
A system has no solution when its equations contradict one another. For example, two equations may describe distinct parallel lines that can never intersect.
Infinitely Many Solutions
A system has infinitely many solutions when its equations are dependent and do not provide enough independent constraints to determine every variable. In two dimensions, this can occur when both equations describe the same line.
How the Solver Calculates the Result
The calculator uses Gauss-Jordan elimination with partial pivoting. It scales the equations, selects suitable pivot coefficients, reduces the augmented matrix, and compares its rank with the number of variables to classify the solution.
Decimal and Negative Coefficients
Coefficients and constants may be positive, negative, zero, or decimal values. Results are rounded to a practical number of significant digits to reduce floating-point noise while preserving very small valid answers.
How Is the Result Calculated?
2 × 2: a₁x + b₁y = c₁; a₂x + b₂y = c₂. 3 × 3: a₁x + b₁y + c₁z = d₁; a₂x + b₂y + c₂z = d₂; a₃x + b₃y + c₃z = d₃.
2 × 2 System of Equations Example
Solve x + y = 5 and x - y = 1. For the first equation, enter 1 for x, 1 for y, and 5 as the constant. For the second equation, enter 1 for x, -1 for y, and 1 as the constant. The solver returns x = 3 and y = 2.
Frequently Asked Questions
How do you solve a system of equations?
You can eliminate variables by combining equations until each variable can be determined. This calculator performs that process automatically using Gauss-Jordan elimination.
Can this calculator solve a 2 × 2 system?
Yes. Select 2 equations and 2 variables, then enter the coefficients of x and y and the constant for both equations.
Can this calculator solve a 3 × 3 system?
Yes. Select 3 equations and 3 variables, then enter the coefficients of x, y, and z and the constant for all three equations.
What should I enter when a variable is missing?
Enter 0 as its coefficient. For example, x + z = 5 should be entered with 0 as the coefficient of y.
Can I enter negative coefficients?
Yes. Enter the minus sign with the coefficient or constant, such as -3 for the y coefficient in 2x - 3y = 7.
Can I use decimal coefficients?
Yes. The solver accepts integer and decimal coefficients and constants.
What does a unique solution mean?
It means exactly one set of values for the variables satisfies every equation in the system.
Why does the system have no solution?
The equations are inconsistent, meaning they impose contradictory conditions and no set of variable values can satisfy all of them.
Why does the system have infinitely many solutions?
The equations are dependent and do not provide enough independent information to determine one value for every variable.
Does the order of the equations matter?
No. Changing the order of the equations does not change the mathematical solution.
What method does the calculator use?
It uses Gauss-Jordan elimination with partial pivoting and equation scaling to solve the augmented matrix and classify the system.
Is the System of Equations Solver free?
Yes. The calculator is free to use and does not require an account.