Paritian

Mathematics

System of Equations Calculator

Any number of simultaneous linear equations solved by Gaussian elimination, with the answer substituted back.

Results

The unknowns x 2 y 3 z -1
Is there exactly one solution? Yes
x 2.0000000000
y 3.0000000000
z -1.0000000000
Determinant -1.00000000
Largest mismatch when substituted back 8.88178e-16
After elimination, in triangular form -3 -1 2 -11 0 1.666667 0.666667 4.333333 0 0 0.200000 -0.200000
Equations 3

What this tool does

Two equations in two unknowns can be done by hand; five in five cannot, and the method that scales is elimination — use each equation in turn to remove one unknown from all the others, until the last one gives an answer directly and the rest fall out backwards. Write your system as an augmented matrix, one equation per line with the constant last, and this page solves it at any size. It also substitutes the answer back into every original equation and reports the largest discrepancy, and it says plainly when the system has no unique solution rather than returning a number anyway.

Formula

[A | b] → [U | c] · xᵢ = (cᵢ − Σⱼ₌ᵢ₊₁ⁿ uᵢⱼ xⱼ) ÷ uᵢᵢ

Variables

SymbolMeaningUnit
aOne equation per line: the coefficients, then the constant—
SOThe unknowns—
UQIs there exactly one solution?—
Xx—
Yy—
Zz—
DDeterminant—
RSLargest mismatch when substituted back—
TUAfter elimination, in triangular form—
NEEquations—

Worked example

  • One equation per line: the coefficients, then the constant2, 1, -1, 8 -3, -1, 2, -11 -2, 1, 2, -3
  • The unknownsx 2 y 3 z -1
  • Is there exactly one solution?Yes
  • x2.0000000000
  • y3.0000000000
  • z-1.0000000000
  • Determinant-1.00000000
  • Largest mismatch when substituted back8.88178e-16
  • After elimination, in triangular form-3 -1 2 -11 0 1.666667 0.666667 4.333333 0 0 0.200000 -0.200000
  • Equations3

Limitations

  • The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.