2 C12H22O8S3 + 33 O2 --> 22 H2O + 24 CO2 + 6 SO2

Neither charges nor an added equation were required.

Number of reactants: 2

Number of Coefficients: 5

a b c d e
At C12H22O8S3 O2 H2O CO2 SO2
H 22 0 -2 0 0
C 12 0 0 -1 0
O 8 2 -1 -2 -2
S 3 0 0 0 -1
Arbitrarily set the first coefficient = 1. This is the smallest value it can take.

a b c d e
1 0 0 0 0 1
22 0 -2 0 0 0
12 0 0 -1 0 0
8 2 -1 -2 -2 0
3 0 0 0 -1 0
By making a series of linear combinations, we combine the equations so that only one of them has a unit value in the left-most position. We continue making linear combinations until there is a stair-step of unit values as shown below.

a b c d e
1 0 0 0 0 1
0 1 -.5 -1 -1 -4
0 0 1 0 0 11
0 0 0 1 0 12
0 0 0 0 1 3
This leads, in this case,to the following trial solutions:

a = 1; b = 16.5; c = 11; d = 12; e = 3

By convention, the smallest set of whole numbers is used for coefficients. The trial coefficients are multiplied by increasing small whole numbers until each number itself becomes a whole number.

a = 2; b = 33; c = 22; d = 24; e = 6

This leads to the balanced chemical equation show in the top line.