Perhaps the most obvious choice of substitutions would be to take
[tex]\begin{cases}u(x,y)=x-8y\\v(x,y)=7x-y\end{cases}\implies\begin{cases}x(u,v)=\dfrac{8v-u}{55}\\\\y(u,v)=\dfrac{v-7u}{55}\end{cases}[/tex]
with Jacobian
[tex]J=\dfrac{\partial(x,y)}{\partial(u,v)}=\begin{bmatrix}-\dfrac1{55}&\dfrac8{55}\\\\-\dfrac7{55}&\dfrac1{55}\end{bmatrix}[/tex]
which has determinant [tex]|J|=\dfrac1{55}[/tex]. Then the integral, assuming it's originally supposed to be
[tex]\displaystyle\iint_R(6x-8y)(7x-y)\,\mathrm dA[/tex]
becomes
[tex]\displaystyle\frac1{55}\int_{u=0}^{u=7}\int_{v=8}^{v=9}\left(\frac{8v-u}{11}-u\right)v\,\mathrm dv\,\mathrm du=\dfrac{931}{363}[/tex]
