How do we balance the chemical equation and give an integer solution?

Question

$$\require{mhchem}\ce{NaOH + H2SO4 -> Na2SO4 + H2O}$$

I'm trying all my best to come up with one solution but cant. in giving the integer solution to this chemical equation is it compulsory?

Answer 1

Denote the numbers on the left with $a$ and $b$ and on the right with $c$ and $d$.

This leads to the system

$$a=2c$$ $$b=c$$ $$a+2b=2d$$ $$a+4b=4c+d$$

considering the number of Na,S,H and O-atoms.

This can also be written as

$\pmatrix {1&0&-2&0\\0&1&-1&0\\1&2&0&-2\\1&4&-4&-1} \pmatrix{a\\b\\c\\d}=\pmatrix{0\\0\\0\\0}$

The kernel of the matrix is $t\pmatrix{2\\1\\1\\2}$, so $(2,1,1,2)$ is the solution.

An easier way to solve the system is to simply start with $b=1$. Then you easily get the solution. If fractions would occur, this would also be no problem because any multiple of a solution is again a solution.