Solve three equations using determinants, with clear steps. Export results instantly for reports, homework, and engineering checks.
Try this system to see how Cramer’s Rule behaves.
| Equation | x | y | z | Constant |
|---|---|---|---|---|
| 1 | 2 | -1 | 3 | 9 |
| 2 | 1 | 4 | -2 | -1 |
| 3 | 3 | -2 | 1 | 4 |
Expected outcome: a unique solution with nonzero D.
Write the system as:
If the main determinant D is nonzero, then:
Compute these 3×3 determinants:
When D = 0, Cramer’s Rule cannot produce a unique solution.
Three‑variable linear systems appear in balance equations, geometry, and multi‑constraint planning. A 3×3 model can represent three unknowns constrained by three independent relations. When the coefficients are well‑posed, the system has a single intersection point. This calculator focuses on fast, transparent solving for x, y, and z.
A 3×3 system can be written as A·u = b, where A is the coefficient matrix, u = [x y z]ᵀ, and b is the constants. The determinant det(A) shows whether A is invertible. If det(A) is zero, equations are dependent or inconsistent, so no solution exists.
Cramer’s Rule replaces one column of A at a time with b to form three new matrices. Their determinants are Dx, Dy, and Dz. When D = det(A) is nonzero, the solution follows: x = Dx/D, y = Dy/D, and z = Dz/D. The method is easy to audit quickly.
The determinants also provide diagnostics. A very small |D| suggests the system is close to singular, meaning tiny input changes may shift the solution dramatically. Comparing Dx, Dy, and Dz to D helps detect scaling issues and reveals which variable is most sensitive. Reviewing these values supports better engineering and classroom interpretation.
Real inputs can span different magnitudes, such as 0.001 and 5000 in the same model. Large spreads can amplify rounding. If numbers are extreme, scale equations by a common factor to keep coefficients similar in size. Then re‑solve and compare. This improves numerical behavior without changing the solution.
After computing x, y, and z, verify by substituting back into the original equations. The left‑hand side should match each constant within your chosen decimal precision. If the mismatch is large, recheck signs, coefficient placement, and decimal points. Systematic validation is especially important when you transpose rows or columns.
Practical uses include mixing problems, force equilibrium in statics, current loops in circuit analysis, and fitting three parameters to three measurements. In project estimating, three constraints might represent budget, labor, and materials simultaneously. In analytics, small systems can appear inside larger optimization workflows as local solves or checkpoints.
Professional reporting benefits from consistent, shareable results. Exporting CSV captures coefficients, determinants, and the computed solution for spreadsheets, audits, and documentation. The PDF export provides a clean snapshot for submissions and site records. Always include the original equations, the D status message, and the formatted x, y, z values.
It means the coefficient matrix is singular. The system may have infinitely many solutions or none at all. Cramer’s Rule cannot produce a single, unique (x, y, z) in that case.
Yes. Enter negatives normally, and type fractions as decimals (for example, 1/3 as 0.3333). The solver accepts scientific notation such as 2.5e-3 as well.
Decimal places only change rounding in the displayed output. The internal calculation uses floating‑point arithmetic. If results shift noticeably, your system may be near singular or poorly scaled.
Substitute x, y, and z back into each equation and recompute the left side. It should match the constant within your selected precision. Large differences usually indicate an input mistake.
Scale each equation by a convenient factor so the coefficients are of similar magnitude. This reduces rounding sensitivity and can produce more stable results when numbers differ by many orders.
The CSV stores all coefficients, constants, determinants (D, Dx, Dy, Dz), the solution values, and the status message. It is useful for spreadsheets, audits, and repeatable reporting.
The PDF captures the system, determinants, status, and the final values in a single page. It is designed for sharing, printing, and attaching to assignments or project documentation.
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.