Question

1. Find the maximum and minimum values of z = 2x + 3y subject to the following constraints


`0 \leqslant x \leqslant 7`

`y \geqslant 1`

`2x - y \geqslant - 5`

`x + y \leqslant 11`
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when x =
and y =
minimum value =
when x =
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Answer 1

Minimum Values:

x = 0, y = 1, z = 3

Maximum Values:

x = 7 , y = 11 , z = 47

Explanation:

FOR MINIMUM VALUES:

0<= x <= 7

It is clear from the above inequality that the minimum value of x must be 0.

y>= 1

This inequality shows that the minimum value of y must be 1. Using these minimum values of x and y to get the minimum value of z:

z = 2x + 3y

z = (2)(0) + (3)(1)

z = 3 (Minimum)

FOR MAXIMUM VALUES:

0<= x <= 7

It is clear from the above inequality that the maximum value of x must be 7.

x + y <= 11

To calculate maximum value of y we can use the minimum value of x in this inequality:

0 + y <= 11

y <= 11

Hence, the maximum value of y is 11.

Using these maximum values of x and y to get the maximum value of z:

z = 2x + 3y

z = (2)(7) + (3)(11)

z = 47 (Maximum)