Reduction from SUBSET-SUM to 0-1-INT-PROG
Let $\langle S, k \rangle$ be an instance of subset sum, where $S=\{a_1, a_2, \dots, a_n\}$. Construct an $0$-$1$-integer program with $n$ binary variables $x_1, \dots, x_n$ and the following constraints:
$$
\begin{align}
a_1x_1+a_2 x_2+ \dots+a_n x_n &amp;\le k \\
-a_1x_1-a_2 x_2 - \dots-a_n x_n &amp;\le -k 
\end{align}
$$
Or, in matrix form:
$A =
\begin{bmatrix}
a_1 &amp; a_2 &amp; \dots &amp; a_n \\
-a_1 &amp; -a_2 &amp; \dots &amp; -a_n
\end{bmatrix}$ and $b=\begin{bmatrix} k \\ -k\end{bmatrix}$.
If you have a solution $S' \subseteq S$ to the subset sum instance, then you can pick $x$ as the characteristic vector of $S'$ to satisfy the constraints (as you correctly point out in your question). I.e., set $x_i = 1$ if and only if $a_i \in S'$ so that $Ax = \begin{bmatrix} k \\ -k\end{bmatrix} = b$.
On the other hand, if $Ax \le b$ then the first constraint is $a_1 + a_2 + \dots + a_n \le k$ and second constraint implies $a_1 + a_2 + \dots + a_n \ge k$. Then, we necessarily have $a_1 + a_2 + \dots + a_n = k$, showing that $S' = \{ a_i \mid x_i = 1\}$ is a solution to the subset sum instance.