# Question: The formula for the line passing through 2 4 3

The formula for the line passing through (2, 4, 3) and (4, 2, 4) in Fig. 5.2 can be written as

(2, 4, 3) + α [(4, 2, 4) – (2, 4, 3)] = (2, 4, 3) + α (2, –2, 1), where 0 ≤ α ≤ 1 for just the line segment between these points. After augmenting with the slack variables x4, x5, x6, x7 for the respective functional constraints, this formula becomes (2, 4, 3, 2, 0, 0, 0) + α (2, – 2, 1, –2, 2, 0, 0). Use this formula directly to answer each of the following questions, and thereby relate the algebra and geometry of the simplex method as it goes through one iteration in moving from (2, 4, 3) to (4, 2, 4). (You are given the information that it is moving along this line segment.)

(a) What is the entering basic variable?

(b) What is the leaving basic variable?

(2, 4, 3) + α [(4, 2, 4) – (2, 4, 3)] = (2, 4, 3) + α (2, –2, 1), where 0 ≤ α ≤ 1 for just the line segment between these points. After augmenting with the slack variables x4, x5, x6, x7 for the respective functional constraints, this formula becomes (2, 4, 3, 2, 0, 0, 0) + α (2, – 2, 1, –2, 2, 0, 0). Use this formula directly to answer each of the following questions, and thereby relate the algebra and geometry of the simplex method as it goes through one iteration in moving from (2, 4, 3) to (4, 2, 4). (You are given the information that it is moving along this line segment.)

(a) What is the entering basic variable?

(b) What is the leaving basic variable?

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