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

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

$"Coefficient " a_1$

$"Coefficient " a_2$

$"Coefficient " a_3$

$"Coefficient " b_1$

$"Coefficient " b_2$

$"Coefficient " b_3$

$"Coefficient " c_1$

$"Coefficient " c_2$

$"Coefficient " c_3$

$"Solution " d_1$

$"Solution " d_2$

$"Solution " d_3$

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:

$z= D_z/D$

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

Inputs

Derivation

See also