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`

Inputs

• `a_1` - the coefficient of the x term in the first equation
• `b_1` - the coefficient of the y term in the first equation
• `c_1` - the coefficient of the z term in the first equation
• `d_1` - the solution term in the first equation
• `a_2` - the coefficient of the x term in the second equation
• `b_2` - the coefficient of the y term in the second equation
• `c_2` - the coefficient of the z term in the second equation
• `d_2` - the solution term in the second equation
• `a_3` - the coefficient of the x term in the third equation
• `b_3` - the coefficient of the y term in the third equation
• `c_3` - the coefficient of the z term in the third equation
• `d_3` - the solution term in the third equation
  

Derivation

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`

See also

Cramer's Rule (three equations)

Cramer's Rule (three equations, solved for x)

Cramer's Rule (three equations, solved for y)

Cramer's Rule Calculator

Determinant of 3-by-3 Matrix