Answer
minimum $z=3$ at $(1,0)$.
Work Step by Step
1. See graph for $\begin{cases}x\ge0\\y\ge0\\x+y\ge1\\3x+2y\le12\\ x+3y\le12 \end{cases}$
2. Identify the corner(s) as $\left(0,1\right),\left(1,0\right)\left(4,0\right),\left(0,4\right),\left(\frac{12}{7},\frac{24}{7}\right)$
3. Evaluate the expression $z=3x+5y$ at the above corners as $z=5,3,12,20,\frac{156}{7}$
4. Thus the minimum is $z=3$ at $(1,0)$.
