  We call the family of free-by-cyclic groups defined by $G = \left< a, t, b_1,
b_2, \ldots b_k \mid at = ta, b_1^{-1}tb_1 = a^{n_1}t, \ldots b_k^{-1}tb_k =
a^{n_k}t \right>$ for $n_1, n_2, \ldots n_k \in \mathbb Z$ linearly mismatched
since the automorphisms used to define the HNN extensions grow linearly at
different rates. Using techniques from Elder's thesis, namely words with a
parallel stable letter structure, we prove that linearly mismatched
free-by-cyclic groups are asynchronously automatic, and thus they have a
solvable word problem.
