I have been learning Linear Programming, and at the moment, the Simplex Method. I have been looking at this resource.
I understand all the steps up until the last statement (page 5), where we find the max $x_1$ and $x_2$. I understand why $Z = 90$ and $x_2 = 30$ but it bewilders me why $x_1 = 0$ instead of $20$.
Any ideas or pointers are super helpful. Thanks!!
Disclaimer: I too am learning the Simplex algorithm.
According to Page 4, Step 4, the new basis set is $\{S_1, x_2\}$. If you understand the derivation of ${\bf{}X}_b$ on Page 5, Step 2, you can see that $S_1 = 20$ and $x_2 = 30$.
Let's back up: The basis set ${\bf{}X}_b$ on Page 4 began as $\{S_1, S_2\}$. Steps 3 and 2, respectively, determined that $S_2$ would be leaving the basis set, while $x_2$ would be entering. That brought us to ${\bf{}X}_b$ = $\{S_1, x_2\}$.
If a variable does not appear in the basis set, its value is $0$ (necessarily, I think). Since $x_1$ is not in the basis set, its value must be $0$.
To verify that $x_1 = 0$, consider from the augmented form $x_1 + x_2 + S_1 = 50$ at the top of Page 4 that $x_1 = 50 - x_2 - S_1$, so $x_1 = 50 - 30 - 20 = 0$.