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Warm Up

Learn how linear programming can help businesses maximize profit or minimize costs by solving systems of linear equations with constraints. Explore examples and find maximum and minimum values of functions.

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Warm Up

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  1. Warm Up Solve the system: 2x + y – 3z = -6 x – y + 2z = 5 3x + 2y – z = 4

  2. Linear Programming Businesses use linear programming to find out how to maximize profit or minimize costs. Most have constraints on what they can use or buy.

  3. Find the minimum and maximumvalue of the function f(x, y) = 3x - 2y. We are given the constraints: • y ≥ 2 • 1 ≤ x ≤5 • y ≤ x + 3

  4. Linear Programming • Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed. • Substitute the vertices into the function and find the largest and smallest values.

  5. 1 ≤ x ≤5 8 7 6 5 4 y ≥ 2 3 2 y ≤ x + 3 1 3 5 4 1 2

  6. Linear Programming • The vertices of the quadrilateral formed are: (1, 2) (1, 4) (5, 2) (5, 8) • Plug these points into the function f(x, y) = 3x - 2y

  7. Linear Programming f(x, y) = 3x - 2y • f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1 • f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5 • f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11 • f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1

  8. Linear Programming • f(1, 4) = -5 minimum • f(5, 2) = 11 maximum

  9. Find the minimum and maximum value of the function f(x, y) = 4x + 3y We are given the constraints: • y ≥ -x + 2 • y ≤ x + 2 • y ≥ 2x -5

  10. y ≥ 2x -5 6 5 4 3 y ≥ -x + 2 2 1 1 2 3 4 5

  11. Vertices f(x, y) = 4x + 3y • f(0, 2) = 4(0) + 3(2) = 6 • f(4, 3) = 4(4) + 3(3) = 25 • f( , - ) = 4( ) + 3(- ) = -1 =

  12. Linear Programming • f(0, 2) = 6 minimum • f(4, 3) = 25 maximum

  13. You Try….

  14. Linear Programing Word Problems

  15. 1. You are going to make and sell bread. A loaf of Irish soda bread is made with 2 c flour and ¼ c sugar. Banana nut bread is made with 4 c flour and 1 c sugar. You will make a profit of $1.50 on each loaf of Irish soda bread and a profit of $4 on each Banana nut. You have 16 c flour and 3 c sugar. How many of each kind of bread should you make to maximize the profit? What is the maximum profit?

  16. 2. A factory produces short-sleeved and long-sleeved shirts. A short-sleeved shirt requires 30 minutes of labor, a long-sleeved shirt requires 45 minutes of labor, and 240 hours of labor are available per day. The maximum number of shirts that can be packaged in a day is 400, so no more than 400 shirts should be produced each day. If the profits on a short-sleeved shirt and a long-sleeved shirt are $11 and $16, respectively, find the maximum possible daily profit.

  17. 3. A nutrition center sells health food to mountain-climbing teams. The Trailblazer mix package contains one pound of corn cereal mixed with four pounds of wheat cereal and sells for $9.75. The Frontier mix package contains two pounds of corn cereal mixed with three pounds of wheat cereal and sells for $9.50. The center has available 60 pounds of corn cereal and 120 pounds of wheat cereal. How many packages of each mix should the center sell to maximize its income?

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