Discuss whether the rule of Cramer can or cannot be used for the following system of equations. Find the solution if possible. $$$\left\{ \begin{array} {rcl} x-y+3z & = & 26 \\ 3x+5y-z &=& 18\\ x+2y-2z &=& -9\end{array}\right.$$$
Development:
The system has the same number of unknowns and equations. Let's look now to see if the determinant of the matrix of the coefficients is other than zero: $$$\Delta=\left| \begin{matrix} 1 & -1 & 3\\ 3 & 5 & -1\\ 1 & 2 & -2 \end{matrix} \right|=-10\neq0$$$
Therefore we can use the rule of Cramer to find the solution. Then, the determinants are: $$\Delta_1=\left| \begin{matrix} 26 & -1 & 3\\ 18 & 5 & -1 \\ -9 & 2 & -2 \end{matrix} \right|=-10, \ \ $$ $$\Delta_2=\left| \begin{matrix} 1 & 26 & 3\\ 3 & 18 & -1 \\ 1 & -9 & -2 \end{matrix} \right|=-50, \ \ $$ $$\Delta_3=\left| \begin{matrix} 1 & -1 & 26\\ 3 & 5 & 18 \\ 1 & 2 & -9 \end{matrix} \right|=-100$$ $$$x=1; \ y=5; \ z=10$$$
Solution:
$$$x=1; \ y=5; \ z=10$$$