The problem requires evaluating the double integral:

$\iint_R (x - 3y) \, dA$

where $R$ is the region with vertices $(0,0)$, $(2,1)$, and $(1,2)$. The transformation:

$x = 2u - v, \quad y = u + 2v$

is suggested for evaluation.

Steps to Solve:

1. Find the Jacobian of the transformation to compute the area element $dA$ in terms of $du dv$.
2. Transform the region $R$ into $(u,v)$-coordinates.
3. Rewrite the integral in terms of $u, v$.
4. Evaluate the transformed integral.

Would you like me to go through each step in detail? Let me know how much explanation you need!

Here are five related questions to deepen your understanding:

1. How do you compute the Jacobian determinant for a coordinate transformation?
2. How do you determine the limits of integration in a change of variables?
3. What are some advantages of using transformations in double integrals?
4. How does the transformation affect the shape of the region $R$?
5. How can this method be extended to triple integrals?

Tip: When choosing a transformation, check if it simplifies the region's boundaries to align with coordinate axes.