Linear Equation System Solver
Solve systems of linear equations with Gaussian elimination and partial pivoting, showing each row operation, the determinant, and whether the system has one solution, none or infinitely many.
Last reviewed by the Radiatus Cloud team
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Three outcomes, not one
A system of linear equations does not always have a single solution. It has exactly one when the equations are independent, none when they contradict each other, and infinitely many when one equation carries no information the others do not. Geometrically, two equations in two unknowns are two lines: they cross at a point, they are parallel, or they are the same line. A solver that reports only "no solution" for the last two cases hides the distinction that usually matters, because a dependent system means the model is under determined rather than wrong.
Why pivoting matters
Gaussian elimination divides by the leading coefficient of each row. If that coefficient is zero the step fails, and if it is very small relative to the others the division amplifies rounding error into the result. Partial pivoting swaps in the row with the largest leading coefficient before each elimination step, which costs almost nothing and turns a numerically unstable method into a reliable one. This solver uses it and reports every swap.
Reading the determinant
The determinant of the coefficient matrix is zero exactly when the system does not have a unique solution. A determinant that is not zero but is tiny compared with the size of the coefficients signals an ill conditioned system, where a small change in the inputs produces a large change in the answer. That is a warning about the data rather than about the arithmetic, and it means the solution should not be trusted to many digits.
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Frequently Asked Questions
What do the three outcomes mean?
One solution means the equations are independent and meet at a single point. No solution means they contradict each other. Infinitely many means at least one equation adds nothing the others do not already say, so the system is under determined rather than wrong.
What is partial pivoting?
Swapping in the row with the largest leading coefficient before each elimination step. It avoids dividing by zero and stops small divisors from amplifying rounding error, which turns an unstable method into a reliable one at almost no cost.
What does a zero determinant tell me?
That the system has no unique solution: either none at all or infinitely many. A determinant that is not zero but is very small relative to the coefficients means the system is ill conditioned and the answer is sensitive to tiny changes in the inputs.
How should I write the equations?
One per line, in a form such as 2x + 3y = 12. Variables can have any single letter or word name, terms may appear on either side of the equals sign, and a variable missing from an equation is treated as having a coefficient of zero.
How many equations do I need?
As many as there are unknowns for a unique solution. Fewer leaves the system under determined with infinitely many solutions, and more can still have a solution if the extra equations agree with the others, or none if they do not.
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How to Use
Enter one equation per line and the solver will find the values of every variable.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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