Step 1 :\( P(x, y) = R(x, y) - C(x, y) \)
Step 2 :\( P(x, y) = (-0.25x^2 - 0.2y^2 - 0.2xy + 200x + 150y) - (90x + 90y + 1300) \)
Step 3 :\( P(x, y) = -0.25x^2 - 0.2y^2 - 0.2xy + 200x + 150y - 90x - 90y - 1300 \)
Step 4 :\( P(x, y) = -0.25x^2 - 0.2y^2 - 0.2xy + 110x + 60y - 1300 \)
Step 5 :\( \frac{\partial P}{\partial x} = -0.5x - 0.2y + 110 \)
Step 6 :\( \frac{\partial P}{\partial y} = -0.4y - 0.2x + 60 \)
Step 7 :\( -0.5x - 0.2y + 110 = 0 \)
Step 8 :\( -0.4y - 0.2x + 60 = 0 \)
Step 9 :\( -y - 0.5x + 150 = 0 \)
Step 10 :\( -0.5x - y + 110 = 0 \)
Step 11 :\( -y - 0.5x + 150 = 0 \)
Step 12 :\( -1.5x + 260 = 0 \)
Step 13 :\( x = \frac{260}{1.5} \)
Step 14 :\( x = 173.\overline{3} \)
Step 15 :\( x = 173 \)
Step 16 :\( -0.4y - 0.2(173) + 60 = 0 \)
Step 17 :\( -0.4y - 34.6 + 60 = 0 \)
Step 18 :\( -0.4y + 25.4 = 0 \)
Step 19 :\( y = \frac{25.4}{0.4} \)
Step 20 :\( y = 63.5 \)
Step 21 :\( y = 63 \)
Step 22 :\( \boxed{x = 173, y = 63} \)