Problems from Cramer's rule

Discuss whether the rule of Cramer can or cannot be used for the following system of equations. Find the solution if possible. ${\begin{array}{rcl} x - y + 3 z & = & 26 \\ 3 x + 5 y - z & = & 18 \\ x + 2 y - 2 z & = & - 9 \end{array}$

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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: $Δ = | \begin{matrix} 1 & - 1 & 3 \\ 3 & 5 & - 1 \\ 1 & 2 & - 2 \end{matrix} | = - 10 \neq 0$

Therefore we can use the rule of Cramer to find the solution. Then, the determinants are: $Δ_{1} = | \begin{matrix} 26 & - 1 & 3 \\ 18 & 5 & - 1 \\ - 9 & 2 & - 2 \end{matrix} | = - 10,$ $Δ_{2} = | \begin{matrix} 1 & 26 & 3 \\ 3 & 18 & - 1 \\ 1 & - 9 & - 2 \end{matrix} | = - 50,$ $Δ_{3} = | \begin{matrix} 1 & - 1 & 26 \\ 3 & 5 & 18 \\ 1 & 2 & - 9 \end{matrix} | = - 100$ $x = 1; y = 5; z = 10$

Solution:

$x = 1; y = 5; z = 10$

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