How many gallons of a 51% alcohol solution should be added to a 57% alcohol solution to make 438 gallons of a 55% solution?
I'm assuming you'll be choosing the amount of 51% solution to use and then adding 57% solution until you have 438 gallons.
Let $\displaystyle w$ be the amount (in gallons) of the weaker (51%) solution in the final product and let $\displaystyle s$ be the amount (in gallons) of the stronger solution.
Now we know that we want the final amount to be 438 gallons, so
$\displaystyle w+s=438$. We also know that we want the final product to be 438 gallons of 55% alcohol, so $\displaystyle 0.51\cdot w+0.57\cdot s=0.55\cdot 438$.
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 $\displaystyle s$, then substitute that into the second equation and solve for $\displaystyle w$:
$\displaystyle s=438-w$
$\displaystyle 0.51\cdot w+0.57(438-w)=0.55\cdot 438$
$\displaystyle 0.51\cdot w+249.66 - 0.57\cdot w=240.9$
$\displaystyle 0.51\cdot w - 0.57\cdot w=-8.76$
$\displaystyle (0.51 - 0.57)w=-8.76$
$\displaystyle -0.06\cdot w=-8.76$
$\displaystyle w=146$
Then you just add 57% solution until you have 438 gallons.
Suppose that the amount of soap solution is $\displaystyle P$ gallons. 20% of this is soap, or $\displaystyle P/5$ gallons of pure soap.
You want $\displaystyle P/5$ gallons to make up 8% of the new solution, so divide $\displaystyle P/5$ by 0.08, giving $\displaystyle 5P/2$ total gallons of solution. Then subtract the amount of original soap solution, and $\displaystyle 5P/2 - P = 3P/2$.
Therefore, if the amount of the original soap solution was $\displaystyle P$ gallons, you'd have to add $\displaystyle 3P/2$ gallons of pure water to make an 8% solution.
For example, if you had 100 gallons of soap solution to start, you'd add 150 gallons of pure water.
(MKN, welcome to the forum. In the future, you should post questions like this to a new thread rather than replying to an old one. Good luck with the soap solution! )