Let be the amount (in gallons) of the weaker (51%) solution in the final product and let be the amount (in gallons) of the stronger solution.
Now we know that we want the final amount to be 438 gallons, so
. We also know that we want the final product to be 438 gallons of 55% alcohol, so .
You then have a system of linear equations in two variables. The easiest way to solve this is probably to solve the first equation for , then substitute that into the second equation and solve for :
Then you just add 57% solution until you have 438 gallons.