#### Answer

The hotel cost per day is $\$80$ and the car rental charge per day is $\$60$.

#### Work Step by Step

We assume the cost per day for the hotel is x and the cost per day for the car is y. According to the statement, package $1$ costs $\$360$ and includes $3$ days at the hotel and a rental car for $2$ days.
Thus, the equation becomes:
$3x+2y=360$
Which can be written as:
$ y=\frac{1}{2}\left( 360-3x \right)$
Package $2$ costs $\$500$ and includes $4$ days at the hotel and a rental car for $3$ days
So, the equation becomes:
$4x+3y=500$
Putting the value of $ y=\frac{1}{2}\left( 360-3x \right)$ in the equation $4x+3y=500$ gives:
$\begin{align}
& 4x+\frac{3}{2}\left( 360-3x \right)=500 \\
& 8x+1080-9x=1000 \\
& x=80
\end{align}$
Now, substitute the value of x in the equation $ y=\frac{1}{2}\left( 360-3x \right)$ as given below:
$\begin{align}
& y=\frac{1}{2}\left( 360-3\times 80 \right) \\
& =60
\end{align}$
Hence, the hotel cost per day is $\$80$ and the car rental charge per day is $\$60$.