$(20$pts$)$ Draw the graph and identify the solution type of three LPs below. $($Plot the constraints and determine the feasible region clearly$).$

a$)$ maxz$=4x_1+x_2$

s$.$t$.8x_1+2x_{2}16$

$5x_1+2x_{2}12$

$x_1,x_{2}0$

b$)$ maxz$=-x_1+3x_2$

s$.$t$.x_1-x_{2}4$

$x_1+2x_{2}4$

$x_1,x_{2}0$

c$)$ maxz$=x_1+x_2$

s$.$t$.$

$x_1+x_{2}4$

$x_1-x_{2}5$

$x_1,x_{2}0$