Take
[tex]\begin{cases}u=x-5y\\v=8x-y\end{cases}[/tex]
so that you have
[tex]\begin{cases}\mathbf x(u,v)=\dfrac{-u+5v}{39}\\\\\mathbf y(u,v)=\dfrac{-8u+v}{39}\end{cases}[/tex]
which gives a Jacobian determinant of
[tex]|\det J|=\left|\begin{vmatrix}\mathbf x_u&\mathbf x_v\\\mathbf y_u&\mathbf y_v\end{vmatrix}\right|=\dfrac1{32}[/tex]
So upon transforming the coordinates to the u-v plane, you have (and I'm guessing on what the integrand actually is)
[tex]\displaystyle\iint_R(10x-5y)(8x-y)\,\mathrm dA=\frac1{32}\int_{u=0}^{u=2}\int_{v=5}^{v=10}\frac5{13}(2u+3v)v\,\mathrm dv\,\mathrm du[/tex]
[tex]=\displaystyle\frac5{416}\int_{u=0}^{u=2}\int_{v=5}^{v=10}(2uv+3v^2)\,\mathrm dv\,\mathrm du=\dfrac{2375}{104}[/tex]
