  Let $G$ be a non-trivial torsion free group and
$s(t)=g_{1}t^{\epsilon_{1}}g_{2}t^{\epsilon_{2}} \cdots g_{n}t^{\epsilon_{n}}=1
\; (g_{i} \in G,\ \epsilon_i=\pm 1)$ be an equation over $G$ containing no
blocks of the form $t^{-1}g_{i}t^{-1}, \; g_{i} \in G$. In this paper we show
that $s(t)=1$ has a solution over $G$ provided a single relation on
coefficients of $s(t)$ holds. We also generalize our results to equations
containing higher powers of $t$. The later equations are also related to
Kaplansky zero-divisor conjecture \cite{K}.
