Question $1.$ Consider the linear program:

$mamize3x_1+x_2$

subject $to5x_1+2x_{2}20$

$5x_1+x_{2}15$

$x_1,x_{2}0.$

The optimal solution is $(x_1,x_2)=(2,5),$ with an optimal value of $11.$

a$)$ Write this linear program in standard form.

b$)$ State the optimal solution in terms of the standard form variables as a $4-$array solution. Identify the basic and nonbasic variables for the optimal solution.

c$)$ Suppose we add $$ to the right hand side of the first constraint, so it becomes $5x_1+2x_{2}20+.$ For what range of $$ does the optimal basis remain unchanged?

d$)$ Write the coordinates of all the corner points and points $(1,4),(12,0)$ in the format $(x_1,x_2,s_1,s_2)$ where $s_1$ and $s_2$ are the slack variables associated with constraints $1$ and $2$ respectively. Is $(2,5)$ a neighbor of $(0,0)?$ List all the neighbors of $(2,5).$