Linear Algebra Problem (Chemistry)

Question

$$C_2H_2 + O_2\rightarrow H_2O + CO_2$$

How do I solve this equation problem with linear algebra? How do I take vectors and matrix so what should I do at here? But the answer is: $$2C_2H_6 + 7O_2\rightarrow 6H_2O + 4CO_2$$

Would someone show me this?

Answer 1

The solution is of the form

$$a\,C_2H_2 + b\ O_2\rightarrow c\ H_2O + d\ CO_2$$ leading to the matter conservation equations

$$\begin{cases}2a-d=0,\\2a-2c=0,\\2b-c-2d=0,\end{cases}$$

which is a system of three linear equations in four unknowns.

As the system is homogeneous, the unknowns are indeterminate to a constant factor, that you can choose to obtain integer values.

Here,

$$a=2,b=5,c=2,d=4.$$

Answer 2

This may help(for combustion reactions): $$C_xH_y + \big{(}x+\frac{y}{4} \big{)} O_2\rightarrow xCO_2+\frac{y}{2}H_2O$$