Question

Maximize −4x + 5y + 70 subject to the constraints:

2x + y ≤ 8
x + 3y ≥ 5
x + y ≤ 6
x ≥ 0,
y ≥ 0

a. Fix any constraints, as needed, and then convert the linear programming problem into a system of linear equations.
b. Give a fully labeled initial tableau, and circle the pivot element.

Answer 1

Explanation:

`\text{To maximize -4x + 5y + 70 subject to } \\ \\ 2x + y \le 8 --- (1) \\ \\ x + 3y \ge 5 --- (2) \\ \\ x + y \le 6----(3) \\ \\ x \ge 0, y \ge 0`

`\text{From above equationn (1)} : 2x + y = 8 \\ \\ \text{Divide boths sides by 8} \\ \\ (2x)/(8) + (y)/(8) = (8)/(8)`

`(x)/(4) + (y)/(8) = 1 \\ \\ x = 4; y = 8`

`\text{From above equationn (2)} : x + 3y = 5 \\ \\ \text{Divide boths sides by 5} \\ \\ (x)/(5) + (3y)/(5) = (5)/(5) \\ \\ x = 5; \ y = 1.66`

`\text{From above equation (3)} : x + y = 6 \\ \\ \text{Divide boths sides by 5} \\ \\ (x)/(6) + (y)/(6) = (6)/(6) \\ \\ x = 6; \ y = 6`

`\text{From the image attached below, we can see the representation in the graph}`

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`\text{Now from equation (1) ad (III)} \\ \\ 2x + y = 8 \\ \\ x+y = 6`

`x`
`= 2`

`From : x + y = 6 \\ \\ 2 + y = 6 \\ \\ y = 6-2 \\ \\ y =4`

`\text{From equation (1) and (II) } \\ \\ \ \ 2x + y = 8 \\ - \\ \ \ x + 3y = 5 \\ \\`

`-5y = -2 \\ \\ y = (2)/(5) \ o r\ 0.4 \\ \\ From : 2x+ y = 8 \\ \\ 2x = 8 - (2)/(5) \\ \\ x = ( 8 - (2)/(5) )/(2) \\ \\ x = 3.8`