Math Problem Statement

z equals 10 x plus 10 yz=10x+10y

subject to

7 x plus 9 y7x+9y

greater than or equals≥126126

16 x minus 11 y16x−11y

greater than or equals≥6767

x plus yx+y

less than or equals≤4343

xx

greater than or equals≥0

y

greater than or equals≥0

Solution

It looks like you have a linear programming problem involving constraints on variables xx and yy. The goal is to either maximize or minimize z=10x+10yz = 10x + 10y subject to the given constraints:

Objective Function:

z=10x+10yz = 10x + 10y

Constraints:

  1. 7x+9y1267x + 9y \geq 126
  2. 16x11y6716x - 11y \geq 67
  3. x+y43x + y \leq 43
  4. x0x \geq 0
  5. y0y \geq 0

Steps to Solve:

  1. Graph the Constraints: Plot the constraints on the coordinate plane to determine the feasible region where all conditions are met.
  2. Find the Feasible Region: The feasible region is the area where all constraints overlap.
  3. Identify Corner Points: Find the intersection points of the lines formed by the constraints; these points are potential candidates for the optimal solution.
  4. Evaluate the Objective Function: Plug the coordinates of the corner points into the objective function z=10x+10yz = 10x + 10y to determine the maximum or minimum value.
  5. Determine the Optimal Solution: The point that gives the highest or lowest value of zz is the optimal solution, depending on whether you are maximizing or minimizing.

Would you like me to plot the feasible region and find the optimal solution, or explain how each constraint is represented graphically?

Related Questions:

  1. What is the significance of each constraint in the context of this problem?
  2. How can we determine the intersection points of the constraints analytically?
  3. How does changing the coefficients in the constraints affect the feasible region?
  4. What happens if the feasible region is unbounded?
  5. How can this linear programming approach be applied to real-world problems?

Tip: Always check if the feasible region is bounded and contains the origin, as this impacts the existence of an optimal solution.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Constraints
Objective Functions

Formulas

Objective function: z = 10x + 10y

Theorems

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Suitable Grade Level

Advanced High School / College