Problem $1$: Solve the following linear program using the graphical solution approach. In solving the problem, please do the following:

Label all constraints and the feasible region.

Draw contour lines to represent the objective function.

Illustrate the objective direction. $($Which way are we trying to move the contour line?$)$

Indicate $($a$)$ whether or not the problem is feasible, $($b$)$ whether or not the problem has an optimal solution, and $($c$)$ whether or not the problem is unbounded.

Label all optimal solutions.

If there is a unique optimal solution, compute the variable values and the objective value for this optimal solution. If there are alternative optimal solutions, compute the variable and objective values for three optimal solutions.

min,$4x_1+x_2$

$s.t.\frac{2}{3}{x}_1+2x_{2}200$

$,-x_1+x_{2}50$

$,x_{1}100$

$,10x_1+4x_{2}600$

$,x_1,x_{2}0$