To solve for the slope of the indifference curve: \(- \frac{\Delta y}{\Delta x} = \frac{MU_x}{MU_y} = MRS_c\)
Budget constraint: \(I = P_x(Q_x) + P_y(Q_y)\)
For utility maximization, we want the slope of the price line to be equal to the slope of the utility curve: \(\frac{P_x}{P_y} = \frac{MU_x}{MU_y}\)
Using the Lagrangian: \(\frac{\partial U}{\partial y} = MU_y ; \frac{\partial U}{\partial x} = MU_x\)
To find out what someone would purchase if there is a tax imposed, just change the budget constraint and take the Lagrangian with the new numbers. 
*REMEMBER: relative prices are what influence people's behavior, not necessarily absolute.