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The Cramer's Rule calculator solves a system of simultaneous linear equations in three variables using Cramer's Rule. The equations are of the form aX+ bY + cZ = d. Enter the coefficients for the three equations
INSTRUCTIONS: Enter the coefficients in the matrix:
- a1, b1, c1, d1
- a2, b2, c2, d2
- a3, b3, c3, d3
Linear Equation Solution (X, Y, Z): The calculator returns the [x, y, z] solution to the set of simultaneous equations.
The Math
Given a system of simultaneous equations:
`a_1 * x + b_1 *y + c_1 = d_1`
`a_2 * x + b_2 *y + c_2 = d_2`
`a_3 * x + b_3 *y + c_3 = d_3`
We can represent these three equations in matrix form using a coefficient matrix, as `[[a_1,b_1, c_1],[a_2,b_2, c_2],[a_3,b_3, c_3]] [[x],[y],[z]] = [[d_1],[d_2],[d_3]]`, where we refer to `[[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]` as the coefficient matrix.
Using Cramer's rule we compute the determinant of the coefficient matrix: `D = |[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]| = a_1*(b_2*c3 - b_3*c_2) + b_1*(c_2*a_3 - a_2*c_3) + c_1* (a_2*b_3 - b_2*a_3)`
We also form the `D_x`, `D_y`, and `D_z` determinants as:
`D_x = |[d_1,b_1,c_1],[d_2,b_2,c_2],[d_3,b_3,c_3]|`
`D_y = |[a_1,d_1,c_1],[a_2,d_2,c_2],[a_3,d_3,c_3]|`
`D_z = |[a_1,b_1,d_1],[a_2,b_2,d_2],[a_3,b_3,d_3]|`
Continuing with Cramer's Rule, we compute the values of x and y as:
`x = D_x/D`
`y = D_y/D`
`z = D_z/D`
Matrix Calculators
- Determinant of 3-by-3 Matrix
- Characteristic Polynomial of a 3x3 matrix
- Inverse of a 3x3 Matrix
- Transpose of a 3x3 Matrix
- Trace of a 3x3 Matrix
- Mirror of a 3x3 Matrix
- 3x3 Matrix Characteristics (Trace, Determinant, Inverse, Characteristic Polynomial)
- Product of a 3x3 matrix and a Scalar
- Product of a 3x3 matrix and a 3x1 matrix
- Product of two 3x3 matrices
- Solving 3 Equations with 3 Unknowns
- Cramer's Rule (three equations, solved for x, y and z)
- Cramer's Rule Calculator
