Balance the equation using algebraic method HCl(aq)+ K2Cr2O7 (aq)+C2H5OH(aq)--->CrCl3(aq)+CO2(g)+KCl(aq)+H20 Please help. Thank you?

aHCl(aq) + bK₂Cr₂O₇ (aq) + cC₂H₅OH(aq) → dCrCl₃(aq) + eCO₂(g) + fKCl(aq) + gH₂O(l)

Compare the numbers of atoms of the both sides:
H: a + 6c = 2g …… [1]
Cl: a = 3d + f …… [2]
K: 2b = f …… [3]
Cr: 2b = d …… [4]
O: 7b + c = 2e + g …… [5]
C: 2c = e …… [6]

From [3]:
f = 2b

From [4]:
d = 2b

Substitute f = 2b and d = 2b into [2]:
a = 3(2b) + 2b
a = 8b

Substitute [6] into [5] :
7b + c = 2(2c) + g
3c + g = 7b …… [7]

Substitute a = 8b into [1]:
8b + 6c = 2g
3c - g = -4b …… [8]

[7] + [8]:
6c = 3b
c = (1/2)b

[7] - [8]:
2g = 11b
g = (11/2)b

Substitute c = (1/2)b into [6]:
2 * (1/2)b = e
e = b

Then, a : b : c : d : e : f : g = 8b : b : (1/2)b : 2b : b : 2b : (11/2)b = 16 : 2 : 1 : 4 : 2 : 4 : 11

The balanced equation is:
16HCl(aq) + 2K₂Cr₂O₇ (aq) + C₂H₅OH(aq) → 4CrCl₃(aq) + 2CO₂(g) + 4KCl(aq) + 11H₂O(l)