Solution

Given objective function is

$\text{Max } Z=2x_1-x_2+2x_3$

In the objective function, the positive coefficient of $x_3$ is $2$.

Taking $x_3$ as the entering variable, we apply the minimum ratio test.

From the constraints:

$2x_1+x_2\le 10$

$x_1+2x_2-2x_3\le 20$

$x_1+2x_3\le 5$

Only the third constraint has positive coefficient of $x_3$.

So,

$2x_3=5$

$x_3=\dfrac{5}{2}$

At the first iteration,

$x_1=0,\ x_2=0,\ x_3=\dfrac{5}{2}$

Now,

$Z=2x_1-x_2+2x_3$

$Z=2(0)-0+2\left(\dfrac{5}{2}\right)$

$Z=5$