<s>[INST] Minimum value of a linear program $x$ and $y$ are real numbers such that $ x\ge 0$ and $y \ge 0$.
If $$ x + y \le 5$$
$$ x + 2y \ge 8$$
Then what is the minimum value of $5x+y$?
There is something wrong in my approach.
I write the first inequality as $$ 5 \ge x + y$$
Adding with the second inequality yields, $$5 + x + 2y \ge x + y + 8$$
Or $$ y \ge 3$$ Thus $$ 9y \ge 27$$
Multiplying the second inequality by $5$ we obtain $$ 5x + 10y \ge 40$$ $$9y \ge 27$$
Hence $5x+y \ge 13$
 [/INST] Our conditions give an interior of $\Delta ABC$,where $A(2,3)$, $B(0,5)$ and $C(0,4).$ 
The system $x=0$ and $x+2y=8$ gives $C(0,4)$;
The system $x=0$ and $x+y=5$ gives $B(0,5)$ 
and the system $x+y=5$ and $x+2y=8$ gives $A(2,3).$
Let $f(x,y)=5x+y.$ 
Thus, $$\min_{x\geq0,y\geq0,x+y\leq5,x+2y\geq8}f=\min\{f(0,5),f(0,4),f(2,3)\}=f(0,4)=4.$$
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