To solve a system of three equations with three unknowns using Cramer's rule, we first need to understand the structure of the equations and how to represent them in matrix form. Cramer's rule provides a straightforward method to find the values of the unknowns by using determinants of matrices.

Given a system of equations, we can express it in the form of an augmented matrix. For example, if we have the equations:

\[\begin{align*}-5x - y + 4z &= 4 \\0x + 3y + 6z &= 21 \\x + y + z &= 6\end{align*}\]

We can create the augmented matrix by organizing the coefficients of the variables and the constants:

\[\begin{bmatrix}-5 & -1 & 4 & | & 4 \\0 & 3 & 6 & | & 21 \\1 & 1 & 1 & | & 6\end{bmatrix}\]

In Cramer's rule, we define \(D\) as the determinant of the coefficient matrix, which consists of the coefficients of \(x\), \(y\), and \(z\). If \(D\) is zero, the system has no unique solution. To calculate \(D\), we use the formula for the determinant of a 3x3 matrix:

\[D = a(ei - fh) - b(di - fg) + c(dh - eg)\end{align*}\]

Where \(a\), \(b\), and \(c\) are the elements of the first row, and \(d\), \(e\), \(f\), \(g\), \(h\), and \(i\) are the elements of the remaining rows. After calculating \(D\), we can find \(D_x\), \(D_y\), and \(D_z\) by replacing the respective columns of the coefficient matrix with the constants from the right side of the equations.

For example, to find \(D_x\), we replace the first column (the coefficients of \(x\)) with the constants:

\[D_x = \begin{vmatrix}4 & -1 & 4 \\21 & 3 & 6 \\6 & 1 & 1\end{vmatrix}\end{align*}\]

Similarly, for \(D_y\) and \(D_z\), we replace the second and third columns with the constants, respectively. The solutions for \(x\), \(y\), and \(z\) can then be calculated as follows:

\[x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}\end{align*}\]

After performing the calculations, we find the values of \(x\), \(y\), and \(z\) that satisfy the original system of equations. This method is efficient for solving linear systems and highlights the relationship between algebra and matrix theory.