Find the maximum value of the function for the polygonal convex set determined by the given system of inequalities (picture provided)

- The maximum value is 86 occurs at (8 , 7)

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

∵ 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)

answer: 819.89 miles across america

step-by-step explanation: