# Linear Algebra Problem (Chemistry) [closed]

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

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?

## closed as off-topic by Dietrich Burde, Davide Giraudo, amWhy, Aweygan, tilperMay 8 '17 at 15:32

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• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Aweygan, tilper
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• No need for linear algebra, see here. – Dietrich Burde May 8 '17 at 13:57
• @DietrichBurde: $ax=b$ is linear algebra, and for more complex reactions, the equations may not be that trivial. – Yves Daoust May 8 '17 at 14:32

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,

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

• I'd comment that the first equation is conservation of carbon, the second equation is conservation of hydrogen, and the third equation is conservation of oxygen. – Ian May 8 '17 at 14:30

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

• Do you suggest that they should learn this by heart instead of understanding how balancing a chemical reactiion works? – Did May 15 '17 at 6:32
• No, but what I'm suggesting is that this equation is usable in cases like OP has mentioned @Did – The Dead Legend May 19 '17 at 6:25