### Balance the given chemical equations. When propane burns in oxygen, it produces carbon dioxide and water:

C_{3}H_{8}+O_{2} \rightarrow CO_{2}+H_{2}O

# Solution:

Write equations for each of the elements.

Oxygen: 0a + 2b = 2c + 1d (The values of the coefficients are the number of oxygen atoms in the chemical equation)

Carbon: 3a + 0b = 1c + 0d

Hydrogen: 8a + 0b = 0c + 2d

Rewrite the equations so that all the variables are on the left side.

0a+2b-2c-d=03a+0b-c+0d=0

8a+0b+0c-2d=0

Solve the equations, easiest way is to use a matrix.

\begin{bmatrix}8 & 0 & 0 & -2 & |0\\3 & 0 & -1 & 0 & |0\\0 & 2 & -2 & -1 & |0\end{bmatrix}Put the matrix in reduced row echelon form.

\begin{bmatrix}1 & 0 & 0 & -\frac{1}{4} & |0\\0 & 1 & 0 & -\frac{5}{4} & |0\\0 & 0 & 1 & -\frac{3}{4} & |0\end{bmatrix} (if you do not know how to get to this form, please check this guide.)

Let s_{1} represent a free parameter.

a = \frac{1}{4}s_{1}b = \frac{5}{4}s_{1}

c = \frac{3}{4}s_{1}

d = s_{1}

Because the lowest common denominator is 4, we will let s_{1} = 4.

Thus, we get:

a = 1b = 5

c = 3

d = 4

Our final balanced equation becomes:

C_{3}H_{8}+5O_{2} \rightarrow 3CO_{2}+4H_{2}OSee a mistake? Comment below so that we can fix it!

How did you get b to equal 5/4 * s1? Why does it not equal 1/2 * s1?

The correct answer is 5/4, however, there was a typo in our reduced row matrix, which we fixed. Thank you very much for pointing it out to us! 🙂 To see the full matrix put into the reduced row form, please take a look here: https://goo.gl/jWKsF3