The integrand and given boundary for R suggests we should use
u = x - 4y
v = 5x - y
with 0 ≤ u ≤ 3 and 6 ≤ v ≤ 9.
Then
x = (4v - u)/19
y = (v - 5u)/19
and the Jacobian for this transformation is
[tex]J=\begin{bmatrix}x_u&x_v\\y_u&y_v\end{bmatrix}=\dfrac1{19}\begin{bmatrix}-1&4\\-5&1\end{bmatrix}\implies|\det J|=\left|\dfrac1{19}\right|=\dfrac1{19}[/tex]
The integral is then
[tex]\displaystyle\iint_R 2\frac{x-4y}{5x-y}\,\mathrm dA = \frac1{19} \int_6^9 \int_0^3 \frac{2u}v\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle \frac2{19} \int_6^9 \frac{3^2-0^2}{2v}\,\mathrm dv[/tex]
[tex]=\displaystyle \frac9{19} \int_6^9 \frac{\mathrm dv}v[/tex]
[tex]=\dfrac9{19}(\ln9-\ln6)=\dfrac9{19}\ln\dfrac96=\boxed{\dfrac9{19}\ln\dfrac32}[/tex]
