Answer
$x=50$
$y=60$
$z=30$
Work Step by Step
Let $x$ denote Midnight Mango
Let $y$ denote Tropical Torrent
Let $z$ denote Pineapple Power
$\begin{cases}
8x+6y+2z=820\\
3x+5y+8z=690\\
3x+3y+4z=450
\end{cases}$
Multiplying Equation 2 by -8/3 and adding it to Equation 1, results in new Equation 1
$\begin{cases}
8x+6y+2z=820\\
-8x-\frac{40}{3}y-\frac{64}{3}=-1840\\
-- -- - -- -- --\\
-\frac{22}{3}y-\frac{58}{3}z=-1020
\end{cases}$
Multiplying both sides by 3, $-22y-58z=-3060$.
Multiplying Equation 3 by -1 and adding it to Equation 2, results in new Equation 2.
$\begin{cases}
3x+5y+8z=690\\
-3x-3y-4z=-450\\
-- -- -- -- --\\
2y+4z=240
\end{cases}$
Multiplying the new equation 2 by 11 and adding it to the new equation 1.
$\begin{cases}
-22y-58z=-3060\\
22y+44z=2640\\
-- -- -- -- --\\
-14z=-420
\end{cases}$
Thus, $z=30$. Substituting it back in, $y=60$ and $x=50$
