I tried with a few algebraic manipulations with no success. for the system:
$$ \begin{pmatrix} x_1' \\ x_2' \end{pmatrix} = \begin{pmatrix} 4 & -5 \\ -7 & -3 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} $$
Tried $x_{1}'=x_{2}$ thus from first equation I get $x_{2}=4x_{1}-5x_{2}$ and then replacing the $x_{2}$ in second equation with $\frac{2}{3}x_{1}$ and the result become $y''=-9y$.
This is not the final answer for some reason, They didn't do any example in class and my logic isn't working here...
In $x'=Ax$ the $2\times2$ matrix $A$ has a characteristic polynomial $$\chi_A(\lambda)=\det(A-λI)=λ^2-tr(A)λ+\det(A).$$ By Cayley-Hamilton, $χ_A(A)=0$ and thus also $$ x''-{\rm tr}(A)x'+\det(A)x=χ_A(D)x=χ_A(A)x=0, $$ here $$ 0=x_k''-x_k'-47x_k,~~k=1,2. $$