Mathematically balancing chemical equation yields weird results
As @Ivan Neretin and @Tyberius pointed out, the negative coefficient of a reactant can be flipped to be positive by moving the reactant to the products' side.

So, in my case, the equations become
$$\begin{align}
\ce{11C7H6O3 + C4H6O3 &amp;-&gt; 9C9H8O4}\\
\ce{8C4H6O3 &amp;-&gt; 2C7H6O3 + 9C2H4O2}
\end{align}$$

Now, since the original equation $\ce{C7H6O3 + C4H6O3 -&gt; C9H8O4 + C2H4O2}$ consists of 2 independent equations which can be separately balanced and combined in an arbitrary proportion. See  More than one way of balancing a chemical equation for a more detailed explanation.