A theater group made appearances in two cities. The hotel charge before tax in the second city was $\$ 1500$ higher than in the first. The tax in the first city was $7.5 \%$, and the tax in the second city was $4.5 \%$. The total hotel tax paid for the two cities was $\$ 517.50$. How much was the hotel charge in each city before tax? First city: Second city: $\$$ $\$$ $x$ S $?$

Step 1 ：Let's denote the hotel charge before tax in the first city as \(x\) and in the second city as \(y\).

Step 2 ：From the problem, we know that the hotel charge in the second city was $1500 higher than in the first. So, we can write this as an equation: \(y = x + 1500\).

Step 3 ：We also know that the total tax paid is $517.50, which is 7.5% of \(x\) plus 4.5% of \(y\). We can write this as another equation: \(0.075x + 0.045y = 517.5\).

Step 4 ：Now we have a system of two equations, and we can solve for \(x\) and \(y\).

Step 5 ：The solution to the system of equations is \(x = 3750\) and \(y = 5250\).

Step 6 ：So, the hotel charge before tax in the first city was \(\boxed{3750}\) and in the second city was \(\boxed{5250}\).