The minimum value of $3x+5y$ such that:$3x+5y\le 15$$4x+9y\le 18$$13x+2y\le 2$$x\ge 0,y\ge 0$ is:

We want to minimize $Z=3x+5y$ subject to non-negativity constraints $x\ge 0,y\ge 0$ and other linear constraints.Since $x$ and $y$ are non-negative, the smallest possible value for $x$ and $y$ is 0.At $(0,0)$, $Z=3(0)+5(0)=0$.Check if $(0,0)$ satisfies all constraints:$3(0)+5(0)=0\le 15$ (True)$4(0)+9(0)=0\le 18$ (True)$13(0)+2(0)=0\le 2$ (True)Since $(0,0)$ is in the feasible region and $x,y\ge 0$, no other point can yield a smaller value for $3x+5y$.Thus, the minimum value is 0.