Let $k \in \mathbb{N}$. For each $1 \leq i \leq k$, let $a_i,b_i \in \mathbb{Z}$ and let \begin{align*} L_i: \mathbb{Z} \to \mathbb{Z} \end{align*} be the affine map $n \mapsto a_i n + b_i$. Assume that $a_i \neq 0$ for every $1 \leq i \leq k$ and that the ordered coefficient pairs $(a_i,b_i)$ are pairwise distinct. Define \begin{align*} F: \mathbb{Z} \to \mathbb{Z} \end{align*} by $F(n)=\prod_{i=1}^k L_i(n)$. Assume that the system has no fixed prime divisor, meaning that for every prime $p$ there exists $n \in \mathbb{Z}$ such that $p \nmid F(n)$. For each prime $p$, define \begin{align*} \nu(p) := \#\left\{r \in \mathbb{Z}/p\mathbb{Z} : F(r) \equiv 0 \pmod p\right\}. \end{align*} For $x \geq 2$, assume that $L_i(n)>1$ for every $n \in \mathbb{N}$ with $1 \leq n \leq x$ and every $1 \leq i \leq k$. Then there exists a constant $C_k>0$, depending only on $k$, such that \begin{align*} \#\left\{n \in \mathbb{N} : 1 \leq n \leq x \text{ and } L_1(n),\dots,L_k(n) \text{ are all prime}\right\} \leq C_k \mathfrak{S}(L_1,\dots,L_k)\frac{x}{(\log x)^k}, \end{align*} where the singular series is \begin{align*} \mathfrak{S}(L_1,\dots,L_k) = \prod_p \left(1-\frac{1}{p}\right)^{-k} \left(1-\frac{\nu(p)}{p}\right). \end{align*}