Mathematically balancing chemical equation yields weird results I found this equation while testing my equation balancer:
$$
\ce{a C7H6O3 + b C4H6O3 -&gt; c C9H8O4 + d C2H4O2}
$$

Obviously, $a = b = c = d = 1$ is a solution. However, if we were to bring in some maths:

$
\begin{cases}
7 a + 4b &amp;= 9c + 2d\\
6a + 6b &amp;= 8c + 4d\\
3a + 3b &amp;= 4c + 2d\\
\end{cases}\\
\begin{cases}
a &amp;= \frac{11}{9}\alpha - \frac{2}{9}\beta\\
b &amp;= \frac{1}{9}\alpha + \frac{8}{9}\beta\\
c &amp;= \alpha\\
d &amp;= \beta\\
\end{cases}
$

This leaves us with many ways of balancing the equation:


when $\alpha = 1, \beta = 1$, $a = b = c = d = 1$; $\ce{C7H6O3 + C4H6O3 -&gt; C9H8O4 + C2H4O2}$
when $\alpha = 1, \beta = 2$, $\begin{cases}
a &amp;= 7\\
b &amp;= 17\\
c &amp;= 9\\
d &amp;= 18\\
\end{cases}$; $\ce{7 C7H6O3 + 17 C4H6O3 -&gt; 9 C9H8O4 + 18 C2H4O2}$
when $\alpha = 2, \beta = 1$...


From what I know, this happens when the equation consists of 2 or more independent equations (as is the case here: More than one way of balancing a chemical equation). However, here the two equations are:

$$
\ce{11 C7H6O3 + C4H6O3 -&gt; 9 C9H8O4}
\\
\ce{-2 C7H6O3 + 8 C4H6O3 -&gt; 9 C2H4O2}
$$

What is concerning here is that the second equation has a negative stoichiometric coefficient. My question is, is a negative coefficient acceptable in an equation, and in this case which is the correct way to balance the equation?
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.