x = gallons of the 7% butterfat milk
y = gallons of the 2% butterfat milk
now, we'll be adding "x" gallons of the 7% one, but we know that, is really "x" gallons, but only 7% of it is butterfat, the rest is water and other subtances.
so, in "x" gallons, how many gallons is butterfat only anyway? namely, what is 7% of "x"? well, (7/100) * x, or 0.07x.
likewise, how many gallons of butterfat are in the 2% one? well, (2/100) * "y" or 0.02y.
we know that, whatever "x" and "y" add up to 275 gallons, thus x + y = 275.
we also know that, the butterfat amounts add up to the same butterfat in the 275 gallons, thus
[tex]\bf \begin{array}{lccclll}
&\stackrel{gallons}{amount}&\stackrel{butterfat~\%}{quantity}&\stackrel{butterfat~gallons}{quantity}\\
&------&------&------\\
\textit{7\% butterfat milk}&x&0.07&0.07x\\
\textit{2\% butterfat milk}&y&0.02&0.02y\\
--------&------&------&------\\
mixture&275&0.05&13.75
\end{array}[/tex]
[tex]\bf \begin{cases}
x+y=275\implies \boxed{y}=275-x\\
0.07x+0.02y=13.75\\
----------\\
0.07x+0.02\left( \boxed{275-x} \right)=13.75
\end{cases}
\\\\\\
0.07x-0.02x+5.50=13.75\implies 0.05x=8.25
\\\\\\
x=\cfrac{8.25}{0.05}\implies x=165[/tex]
how much will it be of the 2% milk? well, y = 275 - x.
