In order to solve the system using Cramer's rule, first let's put the system in matrix form:
[tex]\begin{gathered} \begin{bmatrix}{1} & {1} \\ {1} & {-1}\end{bmatrix}\begin{bmatrix} & {x} \\ & {y}\end{bmatrix}=\begin{bmatrix} & {7} \\ & {1}\end{bmatrix}\\ \\ D\cdot X=B \end{gathered}[/tex]The matrix D is the matrix with the coefficients of x and y from each equation.
The determinant of a 2x2 matrix is given by the product of the terms in the main diagonal minus the product of terms in the secondary diagonal:
[tex]|D|=1\cdot(-1)-1\cdot1=-1-1=-2[/tex]The matrix Dx is given by the matrix D after switching the first column with the coefficient matrix B:
[tex]\begin{gathered} Dx=\begin{bmatrix}{7} & {1} \\ {1} & {-1}\end{bmatrix}\\ \\ |Dx|=7\cdot(-1)-1\cdot1=-7-1=-8 \end{gathered}[/tex]And the matrix Dy is given by the matrix D after switching the second column with the coefficient matrix B:
[tex]\begin{gathered} Dy=\begin{bmatrix}{1} & {7} \\ {1} & {1}\end{bmatrix}\\ \\ |Dy|=1\cdot1-7\operatorname{\cdot}1=1-7=-6 \end{gathered}[/tex]Now, solving the system, we have:
[tex]\begin{gathered} x=\frac{|Dx|}{|D|}=\frac{-8}{-2}=4\\ \\ y=\frac{|Dy|}{|D|}=\frac{-6}{-2}=3 \end{gathered}[/tex]