What this calculator does
Cramer's rule solves a linear system with determinants rather than elimination. Each variable is the ratio of two determinants: one with the constants swapped into that variable's column, and the main coefficient determinant.
It is elegant and impractical at scale. For three equations it needs four determinants and works fine, giving x, y, z of 2.000, 3.000 and −1.000 for the default system.
The formula
The coefficient determinant is computed, then three more with the constants column substituted in turn. Each variable is its determinant divided by the main one.
| Term | Meaning |
|---|---|
| Cramer's rule | Solving a linear system as ratios of determinants. |
| Coefficient determinant | The main determinant, which must be non-zero for a unique solution. |
| Singular system | A zero determinant, meaning no unique solution exists. |
The inputs explained
| Field | What to enter |
|---|---|
| a₁ | Coefficient of x in equation 1. |
| b₁ | Coefficient of y in equation 1. |
| c₁ | Coefficient of z in equation 1. |
| d₁ | Constant on the right of equation 1. |
| a₂ | Coefficient of x in equation 2. |
| b₂ | Coefficient of y in equation 2. |
| c₂ | Coefficient of z in equation 2. |
| d₂ | Constant on the right of equation 2. |
| a₃ | Coefficient of x in equation 3. |
| b₃ | Coefficient of y in equation 3. |
| c₃ | Coefficient of z in equation 3. |
| d₃ | Constant on the right of equation 3. |
When to use it
Solving three simultaneous equations
A standard algebra problem that elimination makes tedious.
Finding where three planes meet
Each equation is a plane, and the solution is their common point.
Checking a hand calculation
Elimination by hand is error-prone and this provides a check.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the first constant move the solution?
The same coefficients with three right-hand values.
| First constant d₁ | x, y, z | Determinant D |
|---|---|---|
| d₁ = 6 | -6.000, 7.000, -11.000 | -1.000 |
| d₁ = 8 | 2.000, 3.000, -1.000 | -1.000 |
| d₁ = 10 | 10.000, -1.000, 9.000 | -1.000 |
Questions
What does a zero determinant mean here?
That the three planes do not meet at a single point. They may share a line, be parallel, or form a triangular prism with no common point. Cramer's rule cannot distinguish those cases, only flag that no unique solution exists.
Why is the rule impractical for large systems?
Because the work grows factorially with size. Solving ten equations this way needs eleven determinants of a ten by ten matrix, which is vastly more work than Gaussian elimination on the same system.
Is it numerically stable?
Less so than elimination with pivoting. Determinants of ill-conditioned matrices lose precision, so for computational work elimination or factorisation is generally preferred even at this size.
Why is it still taught?
Because it gives an explicit formula for each variable, which is useful theoretically and in proofs. It also shows clearly why a non-zero determinant is exactly the condition for a unique solution.
For two equations, see the linear system calculator. For the determinant itself, see the determinant calculator.