Answer:
The maximum value is 126 occurs at (9 , 9)
Step-by-step 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
∵ 8x + 2y = 36 ⇒ (1)
∵ -3x + 6y = 27 ⇒ (2)
- Multiply (1) by -3
∴ -24x - 6y = -108 ⇒ (3)
- Add (2) and (3)
∴ -27x = -81 ⇒ divide both sides by -27
∴ x = 3 ⇒ substitute this value in (1)
∴ 8(3) + 2y = 36
∴ 24 + 2y = 36 ⇒ subtract 24 from both sides
∴ 2y = 12 ⇒ ÷ 2
∴ y = 6
- One vertex is (3 , 6)
∵ 8x + 2y = 36 ⇒ (1)
∵ -7x + 5y = -18 ⇒ (2)
- Multiply (1) by 5 and (2) by -2
∴ 40x + 10y = 180 ⇒ (3)
∴ 14x - 10y = 36 ⇒ (4)
- Add (3) and (4)
∴ 54x = 216 ⇒ ÷ 54
∴ x = 4 ⇒ substitute this value in (1)
∴ 8(4) + 2y = 36
∴ 32 + 2y = 36 ⇒ subtract 32 from both sides
∴ 2y = 4 ⇒ ÷ 2
∴ y = 2
- Another vertex is (4 , 2)
∵ -3x + 6y = 27 ⇒ (1)
∵ -7x + 5y = -18 ⇒ (2)
- Multiply (1) by 7 and (2) by -3
∴ -21x + 42y = 189 ⇒ (3)
∴ 21x - 15y = 54 ⇒ (4)
- Add (3) and (4)
∴ 27y = 243 ⇒ ÷ 27
∴ y = 9 ⇒ substitute this value in (1)
∴ -3x + 6(9) = 27
∴ -3x + 54 = 27 ⇒ subtract 54 from both sides
∴ -3x = -27 ⇒ ÷ -3
∴ x = 9
- Another vertex is (9 , 9)
* Now lets substitute them in f(x , y) to find the maximum value
∵ f(x , y) = 9x + 5y
∴ f(3 , 6) = 9(3) + 5(6) = 27 + 30 = 57
∴ f(4 , 2) = 9(4) + 5(2) = 36 + 10 = 46
∴ f(1 , 5) = 9(9) + 5(9) = 81 + 45 = 126
- The maximum value is 126 occurs at (9 , 9)
