The system of linear equations $x + y + z = 2$,$2x + y - z = 3$,and $3x + 2y + kz = 4$ has a unique solution if

If ${x^a}{y^b} = {e^m}$,${x^c}{y^d} = {e^n}$,${\Delta _1} = \left| {\begin{array}{*{20}{c}} m & b \\ n & d \end{array}} \right|$,${\Delta _2} = \left| {\begin{array}{*{20}{c}} a & m \\ c & n \end{array}} \right|$,and ${\Delta _3} = \left| {\begin{array}{*{20}{c}} a & b \\ c & d \end{array}} \right|$,then the values of $x$ and $y$ are respectively:

For $\alpha, \beta \in [0, 2\pi]$ and $\gamma \in [0, \pi)$, consider the system of equations:
$2 \sin \alpha - \cos \beta + 3 \tan \gamma = 3$
$4 \sin \alpha + 2 \cos \beta - 2 \tan \gamma = 2$
$6 \sin \alpha - 3 \cos \beta + \tan \gamma = 9$
Then, which one of the following is true?

The system of equations $4x + y + 2z = 5$, $x - 5y + 3z = 10$, and $9x - 3y + 7z = 20$ has

For $\alpha, \beta \in R$,suppose the system of linear equations $x-y+z=5$,$2x+2y+\alpha z=8$,and $3x-y+4z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of:

Solve the system of linear equations using the matrix method:
$5x + 2y = 3$
$3x + 2y = 5$