What is the best approach to balance this equation? Why I cannot do it efficiently?
This may be a bit dumb method, but sometimes I do use it as it is pretty straightforward and doesn't require any guessing. I setup a few equations and try solving over integers.
 $$\ce{aCO2 + bH2O &lt;--&gt; cC2H2 + dO2}$$

Balancing $\ce{C}$ atoms both sides,
$$a = 2c\\$$
Balancing $\ce{H}$ atoms both sides,
$$2b=2c\\$$
Balancing $\ce{O}$ atoms both sides,

$$2a+b=2d$$

Solving gives this. Not hard to see that $a=4$ does the job.