Following are the set of equations describing the demand and supply of two goods X and Y:
Demand functions: $$X_d = a_1 - b_1P_x + c_1P_y$$
$$Y_d = a_2 - b_2P_y +c_2P_x$$
$a_1,~ a_2,~ b_1,~ b_2,~ c_1,~ c_2$ are positive.
$$X_s = -f_1 + g_1P_x$$
$$Y_s = -f_2 + g_2Py$$
$f_1,~ f_2,~ g_1,~ g_2$ are positive
Government imposed $t\%$ tax on $X$ and allows $s\%$ subsidy on $Y$ such that government budget remains balanced. Find out the required tax rate. (Clearly show How the set of equations change and the equilibrium prices ?)
I replaced $P_x$ by $P_x(1+t)$ in the two demand equations as according to my understanding consumers now face a higher effective price for $X$. But I'm not sure how to incorporate the subsidy and what would be the economic logic behind such a change.