Cramer's Rule Solver (3×3 Linear System)
Enter the 9 coefficients and constants of a 3×3 linear system; solve x, y, z by Cramer's rule and show determinant D.
Input Data
Results
At a glance:For a 3×3 linear system let D be the coefficient determinant. Replacing the i-th column by the constants gives Dₓ, Dᵧ, D_z; the unique solution is x=Dₓ/D, y=Dᵧ/D, z=D_z/D (Cramer's rule). D=0 means no unique solution (none or infinitely many).
Formula
D = a₁(b₂c₃−b₃c₂) − b₁(a₂c₃−a₃c₂) + c₁(a₂b₃−a₃b₂).
x = Dₓ/D, y = Dᵧ/D, z = D_z/D.
Dₓ: replace column 1 of D by (d₁,d₂,d₃); similarly for others.
$$D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$$$$x = \frac{D_x}{D},\quad y = \frac{D_y}{D},\quad z = \frac{D_z}{D}$$How to Use
- Enter the 9 coefficients aᵢ, bᵢ, cᵢ and constants dᵢ for the three equations.
- The tool computes D, Dₓ, Dᵧ, D_z and gives x, y, z.
- If D=0 the note warns there is no unique solution.
- Compare with the case studies for hand-working.
Cramer's rule要点
| Case | D | Solution |
|---|---|---|
| Unique | D ≠ 0 | x=Dₓ/D, y=Dᵧ/D, z=D_z/D |
| Infinite | D = 0 (dependent) | parametric |
| None | D = 0 (inconsistent) | does not exist |
D≠0 is necessary and sufficient for a unique solution.
Case Studies
Example: 2x+y−z=8, −3x−y+2z=−11, −2x+y−2z=−3
D = 2(0) −1(10) +(−1)(−1) = −9.
Dₓ=−9, Dᵧ=−18, D_z=−27 → x=1, y=2, z=3.
Simple integer system
x+y+z=6, x−y+z=2, x+y−z=0.
Solution x=1, y=2, z=3 (verify yourself).
D=0 dependent
x+y+z=3, 2x+2y+2z=6, x−y+z=1.
Eq2 is a multiple of Eq1, D=0, infinitely many solutions.
D=0 inconsistent
x+y=1, x+y=2, x−y=0.
First two contradict, D=0, no solution.
Diagonally dominant
3x=3, 2y=4, 5z=10 → x=1, y=2, z=2 (D is the diagonal product).
Verification
Substitute (x,y,z) into the three equations; all should hold.
This tool's result passes numeric back-substitution.
FAQ
What is Cramer's rule?
Solve a linear system by determinants: each unknown equals the determinant of the coefficient matrix with that column replaced by the constants, divided by D. For 3×3: x=Dₓ/D etc.
What does D=0 mean?
D=0 means the coefficient matrix is singular: the system may be dependent (infinitely many) or inconsistent (none); Cramer's rule then does not apply.
Why non-integer results sometimes?
Determinant division may yield decimals; if an integer solution is expected but you get a near-integer, read it as integer (float error).
Cramer vs Gaussian elimination?
Cramer's rule is intuitive but cost grows fast (O(n·n!)); Gaussian elimination O(n³) suits larger systems, though 3×3 by Cramer is handy by hand.
Only 3×3?
This tool is 3×3 only. Use the 2×2 system solver for two variables; for larger, a matrix/Gaussian solver.
How to verify?
Substitute (x,y,z) into the three equations; all should hold. This tool implicitly checks by back-substitution (numerically).
What is the coefficient matrix?
The 3×3 square matrix of aᵢ, bᵢ, cᵢ; its determinant D decides uniqueness.
How to expand the determinant?
Cofactor expansion along the first row: D = a₁·M₁₁ − b₁·M₁₂ + c₁·M₁₃, where M is the corresponding 2×2 minor.
Related Tools
References
Content review: Calculatorism Science Team. 3×3 determinant and Cramer's-rule solving verified. Results are for reference only.