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?

A chemical balance will always admit the solution with all zero coefficients: if you put nothing in, you get nothing out. Or, if you double the amount you put in, you will get double the output. 
In linear algebra language, this means that your solution cannot be unique. The set of admissible solutions must form a linear subspace. 
The set of admissible solutions will be the kernel of the matrix you wrote down corresponding to the chemical balance (incidentally, I would put negative signs in front of the first two columns since those represent things on the left hand side of the chemical equation and are things that are consumed), and you are looking for a vector in this kernel with all positive coefficients. Therefore it is in fact necessary for the row-reduced matrix to have negative entries: if the row-reduced matrix has only positive value entries, then it is impossible to have a vector with positive entries in its kernel (since a sum of positive numbers must be positive, and not zero). 
