Working through Three-Equation System on paper
Use partial-pivot Gaussian elimination to reach triangular form, back-substitute, and test the resulting triple in every original equation.
Math calculator
Three-Equation System Solver
Solve three linear equations in x, y, and z using elimination with singular-system detection. The calculation trail makes the reported system solution easier to reproduce.
Three-Equation System Solver inputs
Values for this result
Example from start to finish
The sample system solves to x=1, y=2, z=3, which reproduces all three right sides on substitution. This Three-Equation System Solver example can be compared with two-variable system.
When the result is useful
Three-variable systems handle component balances, spatial intersections, allocation models, and simultaneous unknowns.
Reading the calculation
Three independent linear planes meet at one point when their coefficient matrix has nonzero determinant. Row elimination transforms the system without changing its solution set.
A correct system solution is reliable for Three-Equation System only when the chosen model fits the problem.
Boundaries and common traps
A singular matrix can represent no solution or infinitely many solutions. Nearly dependent rows may amplify input rounding.
Review the sign, scale, and unit of system solution after entering x₁ coefficient, y₁ coefficient, z₁ coefficient, right side 1, x₂ coefficient, y₂ coefficient, z₂ coefficient, right side 2, x₃ coefficient, y₃ coefficient, z₃ coefficient and right side 3.
From calculation to usable answer
Source values may arrive in a different order from the form. Map them explicitly to x₁ coefficient, y₁ coefficient, z₁ coefficient, right side 1, x₂ coefficient, y₂ coefficient, z₂ coefficient, right side 2, x₃ coefficient, y₃ coefficient, z₃ coefficient and right side 3, normalize units, and retain enough precision for the next step after system solution.
Altering one coefficient can rotate a plane and sharply move the common point when the system is nearly singular. Watching this response separates a data-entry mistake from an unexpected but valid value.
If x₁ coefficient, y₁ coefficient, z₁ coefficient, right side 1, x₂ coefficient, y₂ coefficient, z₂ coefficient, right side 2, x₃ coefficient, y₃ coefficient, z₃ coefficient and right side 3 are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
A two-equation solver intersects lines in a plane; this page intersects three planes in three-dimensional coordinate space. This page deliberately reports only the former interpretation.
Label the output as system solution in notes or tables. Store x₁ coefficient, y₁ coefficient, z₁ coefficient, right side 1, x₂ coefficient, y₂ coefficient, z₂ coefficient, right side 2, x₃ coefficient, y₃ coefficient, z₃ coefficient and right side 3 beside it when the result will be reused in a later stage.
Questions about Three-Equation System
What is a singular system?
Its coefficient matrix has determinant zero.
Why use pivoting?
It avoids division by a tiny or zero pivot when a better row is available.
How is the answer checked?
Substitute x, y, and z into all three equations.
Can there be infinitely many solutions?
Yes, when the equations are dependent and compatible.
Do decimals reduce accuracy?
Rounded coefficients can affect nearly singular systems.