The 3 Simultaneous Linear Equation 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.

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`

See also

• Cramer's Rule (three equations)
• Cramer's Rule (three equations, solved for y)
• Cramer's Rule (three equations, solved for z)
• Cramer's Rule Calculator
• Determinant of 3-by-3 Matrix
• Determinant of 2-by-2 Matrix
• Minor of a 3x3 Matrix