Build, solve, and compare three-equation systems with confidence. Inspect coefficients, constants, residuals, and determinants easily. Visualize solution changes clearly before sharing engineering model outputs.
Enter three simultaneous linear equations. The calculator solves for X, Y, and Z using determinant-based matrix methods.
The calculator solves a three-variable simultaneous system in matrix form: A × X = B. For coefficients matrix A, unknown vector X, and constants vector B, the solution uses Cramer’s Rule when the determinant of A is non-zero.
Let the system be:
a11X + a12Y + a13Z = b1
a21X + a22Y + a23Z = b2
a31X + a32Y + a33Z = b3
Then:
X = det(Ax) / det(A)
Y = det(Ay) / det(A)
Z = det(Az) / det(A)
Residuals are also checked:
Residual = left-hand side - right-hand side
Small residuals confirm that the solved variables satisfy the original equations accurately.
| Equation | X Coefficient | Y Coefficient | Z Coefficient | Constant |
|---|---|---|---|---|
| Equation 1 | 2 | 1 | -1 | 8 |
| Equation 2 | -3 | -1 | 2 | -11 |
| Equation 3 | -2 | 1 | 2 | -3 |
| Expected solution: X = 2, Y = 3, Z = -1 | ||||
It solves three simultaneous linear equations with three unknowns. It is useful for engineering balances, control models, circuit equations, and other coupled system calculations.
The determinant shows whether the coefficient matrix is invertible. If it equals zero, the system does not have one unique solution, so Cramer-based solving cannot produce a single exact answer.
A zero determinant means the equations are dependent or inconsistent. You may have infinitely many solutions or no valid solution at all, depending on the constants and equation relationships.
Residuals verify accuracy. They measure how closely the solved X, Y, and Z values reproduce each original equation. Values near zero indicate a consistent numerical solution.
Yes. The calculator accepts integers and decimals. This makes it suitable for real engineering models, transfer functions, calibration equations, and scaled macro-style system approximations.
The chart changes the first equation constant over your chosen range. It then recalculates X, Y, and Z for each point, helping you inspect response trends quickly.
No. The same mathematics applies to engineering systems, process models, force balances, network equations, and any three-equation linear model with coupled unknowns.
Rescale coefficients when values differ by very large magnitudes. Better scaling improves interpretability, reduces rounding issues, and makes diagnostics easier to review in technical work.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.