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`y = |[a_1,b_1, d_1],[a_2,b_2, d_2],[a_3, b_3,d_3]| / |[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3, b_3,c_3]|`

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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`

- `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

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)