System of Equations (3×3)

Gaussian elimination for three unknowns

About this calculator

The System of Equations (3×3) Calculator solves three linear equations aᵢx + bᵢy + cᵢz = dᵢ by Gaussian elimination with partial pivoting, carried out in exact 30-digit decimal arithmetic. It computes the exact determinant of the coefficient matrix, and when it is zero compares the rank of A with the rank of the augmented matrix [A|b] to distinguish an inconsistent system (no solution) from a dependent one (infinitely many). The reduced row-echelon form is shown so you can follow the elimination.

How to use the System of Equations (3×3) calculator

  1. Enter the values for your problem into the input fields.
  2. Read the result — it updates instantly as you type.
  3. Check the formula and the visual explanation to follow how the answer was found.
  4. Copy the page URL to share the exact calculation.

Common examples

  • x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 → x = 1, y = 2, z = 3 (det = 7)
  • x + y + z = 1, 2x + 2y + 2z = 5, x + z = 2 → no solution (det = 0, inconsistent)
  • x + y + z = 3, 2x + 2y + 2z = 6, x − y = 0 → infinitely many solutions (dependent equations)

Frequently asked questions

How does the System of Equations (3×3) calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the System of Equations (3×3) calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.