This equation solves a system of simultaneous linear equations in three variables for z using Cramer's Rule.
The three equations solved for here are of the form:
`a_1 * x + b_1 *y + c_1*z = d_1`
`a_2 * x + b_2 *y + c_2*z = d_2`
`a_3 * x + b_3 *y + c_3*z = d_3`
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 then form the `D_z`determinant as:
`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 y as:
`z= D_z/D`
Cramer's Rule (three equations)
Cramer's Rule (three equations, solved for x)