
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 what points corresponds with the maximum z value

To find the point that corresponds with the maximum value of \( z = 10x + 10y \) given the constraints:

1. \( 7x + 9y \geq 126 \)
2. \( 16x - 11y \geq 67 \)
3. \( x + y \leq 43 \)
4. \( x \geq 0 \)
5. \( y \geq 0 \)

We can solve this as a linear programming problem using the method of vertices (corner points). Here's how to proceed:

### Step 1: Identify the corner points
We need to find the intersection points (corner points) of the constraints, which will be candidates for the maximum value of \( z \).

### Step 2: Solve the system of inequalities
1. **Intersection of \( 7x + 9y = 126 \) and \( 16x - 11y = 67 \)**:
   - Solve these two equations simultaneously to find the intersection point.

2. **Intersection of \( 7x + 9y = 126 \) and \( x + y = 43 \)**:
   - Solve these two equations simultaneously to find the intersection point.

3. **Intersection of \( 16x - 11y = 67 \) and \( x + y = 43 \)**:
   - Solve these two equations simultaneously to find the intersection point.

4. **Check the boundary points** where the constraints intersect the \( x \)- and \( y \)-axes.

### Step 3: Evaluate \( z \) at each corner point
Once we have the coordinates of all relevant corner points, we substitute them into the objective function \( z = 10x + 10y \) to find the maximum value.

Let's calculate the intersection points and then evaluate \( z \) at each one.The intersection points of the constraints are as follows:

1. **Intersection of \( 7x + 9y = 126 \) and \( 16x - 11y = 67 \)**: \((x, y) = (9, 7)\)
2. **Intersection of \( 7x + 9y = 126 \) and \( x + y = 43 \)**: \((x, y) = \left(\frac{261}{2}, \frac{-175}{2}\right)\) (Note: This point is not valid as it has negative \( y \), which violates the \( y \geq 0 \) constraint)
3. **Intersection of \( 16x - 11y = 67 \) and \( x + y = 43 \)**: \((x, y) = (20, 23)\)

Next, we'll evaluate the objective function \( z = 10x + 10y \) at the valid corner points to find the maximum \( z \).The objective function \( z = 10x + 10y \) evaluated at the valid points yields the following:

1. At \((x, y) = (9, 7)\), \( z = 160 \).
2. At \((x, y) = (20, 23)\), \( z = 430 \).

### Conclusion:
The point \((20, 23)\) corresponds to the maximum value of \( z \), which is \( 430 \).

Would you like further details or have any questions?

### Additional Questions:
1. How would the solution change if the coefficients in the objective function were different?
2. What happens if we change one of the inequalities, like making \( x + y \geq 43 \)?
3. Can this method be applied if the constraints are non-linear?
4. How do we interpret the results in a real-world scenario?
5. What is the geometric significance of the corner points in linear programming?

### Tip:
When solving linear programming problems, always check if all points meet the non-negativity constraints (\( x \geq 0 \) and \( y \geq 0 \)) before evaluating the objective function.

Learn how to maximize the objective function z = 10x + 10y subject to linear programming constraints. Solve using the corner points method with detailed steps and calculations. Suitable for advanced high school students studying linear programming.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation