Answer
minimum at $(0,2)$ with $z=8$.
Work Step by Step
Graph the given inequalities as shown
$\begin{cases}x\ge0 \\ y\ge0 \\ x+y\ge2 \\ 2x+3y\le12 \\ 3x+y\le12 \end{cases}$
We can identify the corner points $\left(2,0\right),\left(0,2\right),\left(0,4\right),\left(4,0\right),\left(\frac{24}{7},\frac{12}{7}\right)$ which give the values $z=5x+4y=10,8,16,20,24$. We can find the minimum at $(0,2)$ with $z=8$.
