Answer :
Given the following System of equations:
[tex]\begin{cases}-3x+2y=18 \\ -2x-y=5\end{cases}[/tex]You can identify that it has this form:
[tex]\begin{cases}a_1x+b_1y=c_1_{} \\ a_2x+b_2y=c_2\end{cases}[/tex]Where:
[tex]\begin{gathered} a_1=-3 \\ a_2=-2 \\ b_1=2 \\ b_2=-1 \\ c_1=18_{} \\ c_2=5 \end{gathered}[/tex]The determinant D is, by definition:
[tex]D=\begin{bmatrix}{a_1} & {b_1} & {} \\ {a_2} & {b_2} & {} \\ {} & {} & \end{bmatrix}=a_1b_2-a_2b_1[/tex]Then, in this case this is:
[tex]D=\begin{bmatrix}{-3} & {2_{}} & {} \\ {-2_{}} & {-1_{}} & {} \\ {} & {} & \end{bmatrix}=(-3)(-1)-(-2)(2)=7[/tex]By definition, the determinant associated with "x" is given by:
[tex]D_x=\begin{bmatrix}{c_1} & {b_1} & {} \\ {c_2} & {b_2} & {} \\ {} & {} & \end{bmatrix}=c_1b_2-c_2b_1[/tex]Then, in this case:
[tex]D_x=\begin{bmatrix}{18_{}} & {2_{}} & {} \\ {5_{}} & {-1_{}} & {} \\ {} & {} & \end{bmatrix}=(18)(-1)-(5)(2)=-28[/tex]The determinant associated with "y" is given by:
[tex]D_y=\begin{bmatrix}{a_1} & {c_1} & {} \\ {a_2} & {c_2} & {} \\ {} & {} & \end{bmatrix}=a_1c_2-a_2c_1[/tex]Then, this is:
[tex]D_y=\begin{bmatrix}{-3_{}} & {18_{}} & {} \\ {-2_{}} & {5_{}} & {} \\ {} & {} & \end{bmatrix}=(-3)(5)-(-2)(18)=21[/tex]The solution of the System of equations can be found as following:
1. For the x-coordinate:
[tex]x_{}=\frac{D_x}{D}=\frac{-28}{7}=-4[/tex]2. For the y-coordinate:
[tex]y=\frac{D_y}{D}=\frac{21}{7}=3[/tex]The answers are:
[tex]\begin{gathered} D=7 \\ D_x=-28 \\ D_y=21 \\ \text{Solution}=(-4,3) \end{gathered}[/tex]