"the total water of the barrel and the water is 98.4 pounds" should, I think read "the total weight of the barrel and the water is 98.4 pounds."
What are you expected to do here?
Perhaps finding the weight of the barrel is your goal.
In that case, set up a system of linear equations, using b for the weight of the barrel and w for the weight of the water in the barrel.
Then:
1b + 10g = 98.4 lb
1b + 20g = 181.8 lb
Solve for b and g by elimination. Mult. the 1st eqn by -2: -2b - 20g = 196.8
Add that result to the second equation:
-2b - 20g = -196.8
1b + 20g = 181.8 lb
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Then -b = -15 lb. Thus, the weight of the barrel is 15 lb.
What is g, the weight of 1 gallon of water?
Subst. 15 lb for b in the equation 1b + 20g = 181.8 lb
becomes: 15lb + 20g = 181.8 lb
g = 9.09 lb
Thus, the barrel weights 15 lb and each gallono of water weights 9.09 lb.