Enter your system of linear equations as a coefficient matrix and constants vector, then get each variable's solution using Cramer's Rule. Choose your system size (2×2 or 3×3), fill in the coefficients and constants, and the calculator returns the values of x, y (and z for 3×3) along with the main determinant and each substituted determinant.
x
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y
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z (3×3 only)
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Main Determinant (D)
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Determinant Dₓ
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Determinant D_y
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Determinant D_z (3×3 only)
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Cramer's Rule is a mathematical theorem that solves a system of linear equations using determinants. For each variable, you replace the corresponding column in the coefficient matrix with the constants vector, compute that determinant, and divide by the main determinant. It works for any n×n system as long as the main determinant is non-zero.
Cramer's Rule cannot be applied when the main determinant (D) equals zero. A zero determinant means the system is either inconsistent (no solution) or dependent (infinitely many solutions). In both cases, the system cannot be solved uniquely with this method.
Arrange your equations in standard form: place all variable terms on the left and constants on the right. For a 2×2 system, enter the x and y coefficients for each row and the corresponding constant. For a 3×3 system, also include the z coefficients and a third equation row.
The main determinant D is the determinant of the coefficient matrix — the matrix formed only from the variable coefficients, without the constants. It is the denominator used in all of Cramer's Rule calculations. If D ≠ 0, the system has a unique solution.
These are the substituted determinants used to find each variable. Dₓ is computed by replacing the x column of the coefficient matrix with the constants vector. D_y replaces the y column, and D_z replaces the z column. Each variable is then found by dividing its determinant by D.
Yes. You can enter any real number as a coefficient or constant, including negative values and decimals. The calculator computes determinants using standard arithmetic, so negatives and non-integers are handled correctly.
Cramer's Rule is elegant for small systems (2×2 or 3×3) and is commonly used in textbooks and engineering problems. However, for larger systems it becomes computationally expensive compared to methods like Gaussian elimination, so it is best suited to the smaller systems this calculator supports.
If the main determinant D equals zero, the calculator will flag that the system has no unique solution. This means the equations are either contradictory (no solution exists) or redundant (infinitely many solutions exist), and Cramer's Rule cannot be applied.