Balancing an acid-base chemical reaction I tried balancing this chemical equation 
$$\mathrm{Al(OH)_3  + H_2SO_4 \to Al_2(SO_4)_3 + H_2O}$$
with a system of equations, but the answer doesn't seem to map well. I get a negative coefficient which is prohibitive in this equations. How do I interpret the answer?

Let $x$ be the number of $\mathrm{Al(OH)_3}$; $y$ the number of $\mathrm{H_2SO_4}$; $z$ the number of $\mathrm{Al_2(SO_4)_3}$, and $w$ the number of $\mathrm{H_2O}$. Looking at the number of $\mathrm{Al}$, you get $x = 2z$. Looking at $\mathrm{O}$, you get $3x + 4y = 12z + w$. Looking at $\mathrm{H}$ you get $3x + 2y = 2w$; and looking at $\mathrm{S}$ you get $y = 3z$.
That looks like what you are getting from Wolfram, except you have the wrong signs for $z$ and $w$; unless you are interpreting the first two entries to represent the "unknowns", and the last two to represent the "solutions". I would translate into equations the usual way.
What you have is the following system of linear equations:
$$\begin{array}{rcrcrcrcl}
x & & &  -& 2z & & & = & 0\\
3x & + & 4y & - & 12z & - & w & = & 0\\
3x & + & 2y &    &    & - & 2w & = & 0\\
   &   & y & - & 3z & & & = & 0
\end{array}$$
This leads (after either some back-substitution from the first and last equations into the second and third, or some easy row reduction) to $x=2z$, $y=3z$, and $6z=w$. Since you only want positive integer solutions, setting $z=1$ gives $x=2$, $y=3$, and $w=6$, yielding the smallest solution:
$$\mathrm{2 Al(OH)_3 + 3H_2SO_4 \to Al_2(SO_4)_3 + 6H_2O}$$
