Simultaneous Equations Solver
Solved, then checked by substitution.
Solved, then checked by substitution. States the assumption instead of hiding it.
Solution
x = 2, y = 3, z = -1
Verified by substitution — largest residual 8.9e-16
Solved by Gaussian elimination with partial pivoting, then checked by substituting the answer back into the original equations. The largest leftover error is the residual, and it is reported because a system can be solvable in principle and badly behaved in practice. Pivoting is not an optimisation. Without it, a small leading coefficient becomes a divisor and amplifies rounding error through every subsequent row — which is how naive elimination produces confident nonsense on perfectly ordinary systems. This is a better route than computing the inverse and multiplying, both in speed and in accuracy. The inverse is needed far less often than it is used.
How the Simultaneous Equations Solver works
Solves a 2×2 or 3×3 linear system by Gaussian elimination with partial pivoting, then substitutes the answer back and reports the largest leftover error. A system can be solvable in principle and badly behaved in practice.
Also known as: solve two equations two unknowns · three variable system solver · x y z equation solver · gaussian elimination calculator
Frequently asked questions
What if the determinant is zero?
There is no unique solution. Either the equations describe the same relationship more than once — infinitely many solutions — or they contradict each other and there are none. Telling the two apart needs the rank of the augmented matrix.
What does the residual mean?
How far the returned solution is from actually satisfying the original equations. It should be essentially zero; a large residual signals a badly conditioned system whose answer should not be trusted.
What is partial pivoting?
Reordering rows so the largest available coefficient becomes each pivot. Without it, a small pivot amplifies rounding error through the rest of the elimination and produces confident nonsense on ordinary systems.
What does a 3×3 system mean geometrically?
Each equation is a plane, and the solution is where all three meet. A unique solution is a single point; a zero determinant means they meet along a line, coincide, or form a prism with no common point.
Is this better than using the inverse?
Yes, on both speed and accuracy. Computing an inverse in order to multiply by it does more arithmetic and accumulates more error than solving the system directly.
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