Hello, khrst4!
Minimize: .$\displaystyle Z \:= \:3x + 2y$
subject to: .$\displaystyle \begin{Bmatrix}2x + y & \geq & 10 \\
3x + 2y & \leq & 6 \\ x + y & \geq & 6 \end{Bmatrix}$ . and: .$\displaystyle \begin{Bmatrix}x & \geq & 0 \\ y & \geq & 0 \end{Bmatrix}$
(a) Using the Big M method, construct the complete first simplex tableau for the simplex method
and identify the corresponding initial (artificial) BF solution.
Also identify the initial entering basic variable and the leaving basic variable.
(b) Work through the simplex method step by step to solve the problem.
I haven't use the Simplex Method in decades.
. . But I can solve it with traditional graphic methods.
From $\displaystyle x \geq 0,\;y \geq 0$, we are in Quadrant 1.
Graph the line: .$\displaystyle 2x + y \:=\:10$
It has intercepts: $\displaystyle (5,0),\:(0,10)$
Draw the line and shade the region above the line.
Graph the line: .$\displaystyle 3x + 2y \:=\:6$
It has intercepts: $\displaystyle (2,0),\;(0,3)$
Draw the line and shade the region below the line.
Graph the line: .$\displaystyle x + y \:=\:6$
It has intercepts: $\displaystyle (6,0),\;(0,6)$
Draw the line and shade the region above the line. Code:

10 *
*
 *
 *
 *
 *
 *
 *
6 * * *
 * * *::::
 * * *::::::::
 * o::::::::::::
 * *:::::::::::
 * * *::::::::::
3 * * *:::::::::
*  * *::::::::
*  **:::::::
*  o::::::
*  **::::
 *          +          * o  
2  5 6
The shaded region has vertices: .$\displaystyle (6,0),\;(4,2),\;(2,6)$
Test those in the $\displaystyle Z$function for minimum $\displaystyle Z.$