Consider the region given by

Question

Consider the region given by

x≤1

−2 x+y ≥5

x+y−z ≤2

x,y,z ≥0

Write down the basic solution obtained by choosing x,y,z as pivots. Enter in the form (x,y,z,s1,s2,s3)

Answer 1

To find the basic solution obtained by choosing x, y, and z as pivots, we'll start with the following system of inequalities:

1. x ≤ 1
2. -2x + y ≥ 5
3. x + y - z ≤ 2
4. x ≥ 0
5. y ≥ 0
6. z ≥ 0

First, let's rewrite the inequalities in standard form (with variables on the left side and constants on the right side):

1. x - s1 = 1
2. -2x + y + s2 = 5
3. x + y - z + s3 = 2
4. x ≥ 0
5. y ≥ 0
6. z ≥ 0
7. s1 ≥ 0
8. s2 ≥ 0
9. s3 ≥ 0

Now, we'll identify the basic variables and their corresponding values. We choose x, y, and z as pivots, and the remaining variables (s1, s2, s3) will be the slack variables:

Basic Variables: x = 1 y = 0 z = 2

Non-Basic Variables (Slack Variables): s1 = 0 s2 = 5 s3 = 0

So, the basic solution obtained by choosing x, y, and z as pivots is (1, 0, 2, 0, 5, 0).