Minimize z = 8x +10y Subject to
2x +y > 7, 2x + 3y > 15, y> 2, x>0, y> 0
Answers
EXPLANATION.
Minimize : z = 8x + 10y.
⇒ 2x + y ≥ 7. - - - - - (1).
⇒ 2x + 3y ≥ 15. - - - - - (2).
⇒ y ≥ 2. - - - - - (3).
⇒ x > 0 and y > 0.
As we know that,
We can write equation as,
⇒ 2x + y = 7. - - - - - (1).
⇒ 2x + 3y = 15. - - - - - (2).
⇒ y = 2. - - - - - (3).
From equation (1), we get.
⇒ 2x + y = 7. - - - - - (1).
Put the value of x = 0 in the equation, we get.
⇒ 2(0) + y = 7.
⇒ y = 7.
Their Co-ordinates = (0,7).
Put the value of y = 0 in the equation, we get.
⇒ 2x + (0) = 7.
⇒ 2x = 7.
⇒ x = 7/2.
⇒ x = 3.5.
Their Co-ordinates = (3.5,0).
From equation (2), we get.
⇒ 2x + 3y = 15. - - - - - (2).
Put the value of x = 0 in the equation, we get.
⇒ 2(0) + 3y = 15.
⇒ 3y = 15.
⇒ y = 5.
Their Co-ordinates = (0,5).
Put the value of y = 0 in the equation, we get.
⇒ 2x + 3(0) = 15.
⇒ 2x = 15.
⇒ x = 15/2.
⇒ x = 7.5.
Their Co-ordinates = (7.5,0).
From equation (3), we get.
⇒ y = 2. - - - - - (3).
Their Co-ordinates = (0,2).
This three curve intersects at a point = (1.5,4) & (2.5,2) & (4.5,2).
Minimize : z = 8x + 10y.
(1) = Co-ordinates = (1.5,4).
⇒ z = 8(1.5) + 10(4).
⇒ z = 12 + 40.
⇒ z = 52.
(2) = Co-ordinates = (4.5,2).
⇒ z = 8(4.5) + 10(2).
⇒ z = 36 + 20.
⇒ z = 56.
(3) = Co-ordinates = (0,7).
⇒ z = 8(0) + 10(7).
⇒ z = 70.
z is minimum at Co-ordinates = (1.5,4) = 52.
