The unbalanced chemical equation is
"CH"_3"CH"_2"OH" + "O"_2 → "CO"_2 + "H"_2"O"
A method that often works is first to balance everything other than "O" and "H", then balance "O", and finally balance "H".
Another useful procedure often is to start with what looks like the most complicated formula.
The most complicated formula looks like "CH"_3"CH"_2"OH". We put a 1 in front of it to remind ourselves that the number is now fixed.
color(red)(1)"CH"_3"CH"_2"OH" + "O"_2 → "CO"_2 + "H"_2"O"
Balance "C":
We have "2 C" on the left, so we need "2 C" on the right. We put a 2 in front of the "CO"_2.
color(red)(1)"CH"_3"CH"_2"OH" + "O"_2 → color(orange)(2)"CO"_2 + "H"_2"O"
Balance "O":
We can't balance "O" because there are two compounds containing "O" that have no coefficients.
Balance "H":
We have fixed "6 H" on the left. We need "6 H" on the right. Put a 3 in front of "H"_2"O".
color(red)(1)"CH"_3"CH"_2"OH" + "O"_2 → color(orange)(2)"CO"_2 + color(blue)(3)"H"_2"O"
Balance "O"
We have fixed "7 O" atoms on the right and one on the left. We need six more "O" atoms on the left. Put a 3 in front of "O"_2.
color(red)(1)"CH"_3"CH"_2"OH" + color(purple)(3)"O"_2 → color(orange)(2)"CO"_2 + color(blue)(3)"H"_2"O"
Every formula now has a coefficient. We should have a balanced equation.
Let's check.
bb("Atom" color(white)(m)"lhs"color(white)(m)"rhs")
color(white)(m)"C"color(white)(mmm)2color(white)(mml)2
color(white)(m)"H"color(white)(mmm)6color(white)(mml)6
color(white)(m)"O"color(white)(mmm)7color(white)(mml)7
All atoms balance. The balanced equation is
"CH"_3"CH"_2"OH" + "3O"_2 → "2CO"_2 + "3H"_2"O"
