#### Answer

The restaurant should should purchase $6$ two-seat tables and $11$ four-seat tables.

#### Work Step by Step

Let us assume x to be the number of two-seat tables and y to be the number of four-seat tables.
The provided data can be represented in the form of equations as follows:
$4x+2y=56$ (I)
$x+y=17$ (II)
Solve for x in the second equation to obtain its value in terms of y:
$\begin{align}
& x+y=17 \\
& x=17-y
\end{align}$
Put the value of $x$ from the second equation in the first equation:
$\begin{align}
& 4x+2y=56 \\
& 4\cdot \left( 17-y \right)+2y=56
\end{align}$
Simplify:
$\begin{align}
& 68-4y+2y=56 \\
& 68-56=2y \\
& 12=2y
\end{align}$
And divide by $2$ on both sides
$\begin{align}
& y=6 \\
& x=17-y \\
\end{align}$
Put the value $y=6$ in $x=17-y$:
$\begin{align}
& x=17-6 \\
& x=11 \\
\end{align}$
Hence, the number of four-seat tables is $11$ and the number of two-seat tables is $6$.