$$\eqalign{
& {\alpha _{1x}}f + {\alpha _{2x}}g + {\alpha _1}{f_x} + {\alpha _2}{g_x} = 2x \cr
& {\alpha _{1y}}f + {\alpha _{2y}}g + {\alpha _1}{f_y} + {\alpha _2}{g_y} = 0 \cr
& {\beta _{1y}}f + {\beta _{2y}}g + {\beta _1}{f_y} + {\beta _2}{g_y} = 2y \cr
& {\beta _{1x}}f + {\beta _{2x}}g + {\beta _1}{f_x} + {\beta _2}{g_x} = 0 \cr
& {\gamma _{1x}}f + {\gamma _{2x}}g + {\gamma _1}{f_x} + {\gamma _2}{g_x} = y \cr
& {\gamma _{1y}}f + {\gamma _{2y}}g + {\gamma _1}{f_y} + {\gamma _2}{g_y} = x \cr} $$