Answer:
- The maximum value is 86 occurs at (8 , 7)
Explanation:
* Lets remember that a function with 2 variables can written
f(x , y) = ax + by + c
- We can find a maximum or minimum value that a function has for
the points in the polygonal convex set
- Solve the inequalities to find the vertex of the polygon
- Use f(x , y) = ax + by + c to find the maximum value
∵ 3x + 4y = 19 ⇒ (1)
∵ -3x + 7y = 25 ⇒ (2)
- Add (1) and (2)
∴ 11y = 44 ⇒ divide both sides by 11
∴ y = 4 ⇒ substitute this value in (1)
∴ 3x + 4(4) = 19
∴ 3x + 16 = 19 ⇒ subtract 16 from both sides
∴ 3x = 3 ⇒ ÷ 3
∴ x = 1
- One vertex is (1 , 4)
∵ 3x + 4y = 19 ⇒ (1)
∵ -6x + 3y = -27 ⇒ (2)
- Multiply (1) by 2
∴ 6x + 8y = 38 ⇒ (3)
- Add (2) and (3)
∴ 11y = 11 ⇒ ÷ 11
∴ y = 1 ⇒ substitute this value in (1)
∴ 3x + 4(1) = 19
∴ 3x + 4 = 19 ⇒ subtract 4 from both sides
∴ 3x = 15 ⇒ ÷ 3
∴ x = 5
- Another vertex is (5 , 1)
∵ -3x + 7y = 25 ⇒ (1)
∵ -6x + 3y = -27 ⇒ (2)
- Multiply (1) by -2
∴ -6x - 14y = -50 ⇒ (3)
- Add (2) and (3)
∴ -11y = -77 ⇒ ÷ -11
∴ y = 7 ⇒ substitute this value in (1)
∴ -3x + 7(7) = 25
∴ -3x + 49 = 25 ⇒ subtract 49 from both sides
∴ -3x = -24 ⇒ ÷ -3
∴ x = 8
- Another vertex is (8 , 7)
* Now lets substitute them in f(x , y) to find the maximum value
∵ f(x , y) = 2x + 10y
∴ f(1 , 4) = 2(1) + 10(4) = 2 + 40 = 42
∴ f(5 , 1) = 2(5) + 10(1) = 10 + 10 = 20
∴ f(8 , 7) = 2(8) + 10(7) = 16 + 70 = 86
- The maximum value is 86 occurs at (8 , 7)