Min $6$X$1+2$X$2$

St$.-3$X$1+5$X$2<=10(1)$

$5$X$1+6$X$2<=30(2)$

$3$X$1+4$X$2>=12(3)$

$-2$X$1+4$X$2>=0(4)$

X$1,$ X$2>=0$

a$.$ Please use graphical method to find the FEASIBLE REGION. Please label the $4$ constraint lines in your graph. Indicate the points you used to draw the lines.

b$.$ Please draw an objective function line in the graph of a$.$ Mark the optimal solution as a star. List the two constraint boundary lines that are crossed at the optimal solution point here.

c$.$ Please get the optimal solution $($list the system of two equations and solve it with details$)$: How much is X$1$ and X$2?$ You need to indicate the minimized output of the objective function.

d$.$ use Spreadsheet and Solver to get the optimal solution. From your spreadsheet, how much is the optimal solution and the minimized objective function output $($state by referring to the Spreadsheet cells where you acquire information$)?$