Answer
$2$ VitaMax, $1$ Vitron, $2$ VitaPlus
Work Step by Step
Step 1. Establish the system of equations, assuming $x$ VitaMax, $y$ Vitron, and $z$ VitaPlus are needed:
$\begin{cases} 5x+10y+15z=50 \\ 15x+20y+0z=50 \\ 10x+10y+10z=50 \end{cases}$
Step 2. Establish the augmented matrix of the system and use the Gauss Eliminations method:
$\begin{vmatrix} 5 & 10 & 15 & 50 \\15 & 20 & 0 & 50\\10 & 10 & 10 & 50 \end{vmatrix} \begin{array}( R_1/5\to R_1\\R_2/5\to R_2\\R_3/10\to R_3\\ \end{array}$
Step 3. Do the operations given on the right side of the matrix.
$\begin{vmatrix} 1 & 2 & 3 & 10 \\3 & 4 & 0 & 10\\1 & 1 & 1 & 5 \end{vmatrix} \begin{array}( R_1-R_3\to R_1\\R_2-3R_3\to R_2\\.\\ \end{array}$
Step 4. Do the operations given on the right side of the matrix.
$\begin{vmatrix} 0 & 1 & 2 & 5 \\0 & 1 & -3 & -5\\1 & 1 & 1 & 5 \end{vmatrix} \begin{array}( R_1-R_2\to R_1\\.\\.\\ \end{array}$
Step 5. Do the operations given on the right side of the matrix.
$\begin{vmatrix} 0 & 0 & 5 & 10 \\0 & 1 & -3 & -5\\1 & 1 & 1 & 5 \end{vmatrix} \begin{array}( \\.\\.\\ \end{array}$
Step 6. Without rearranging the order of rows, row 1 gives $5z=10$ and $z=2$. Back substitute it to row 2, we get $y=3z-5=1$. And row 3 gives $x=5-y-z=2$
Step 7. The final results are: $2$ VitaMax, $1$ Vitron, and $2$ VitaPlus are needed.
