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Here is the graph of the system of inequalities. The blue line represents the constraint for A grade coffee, \( 4x + 7y = 700 \), while the green line represents the constraint for B grade coffee, \( 8x + 3y = 600 \). The shaded region indicates the feasible set of values for \( x \) (deluxe blend packages) and \( y \) (special blend packages) that satisfy all the constraints.

In this case:
- \( x \geq 0 \) and \( y \geq 0 \), meaning only the first quadrant is relevant.
- The region where the blue and green lines overlap is shaded, representing the possible combinations of deluxe and special blend packages that can be produced with the available coffee.

Would you like further details on how to find the corner points or solve for a particular solution?

Here are 5 related questions for further exploration:
1. How do we find the exact coordinates of the intersection points of the constraints?
2. What is the maximum number of packages of deluxe blend that can be produced?
3. What is the maximum number of packages of special blend that can be produced?
4. How would the solution change if you had more A grade coffee available?
5. What happens if the proportions of A and B grade coffee in each blend change?

**Tip:** In optimization problems, corner points of the feasible region often provide optimal solutions when maximizing or minimizing a linear objective function.

For a given week, Amy’s Coffee House has available 700 ounces of A grade coffee and 600 ounces of B grade coffee. These are blended into packages as follows: a deluxe blend that contains 4 ounces of A grade coffee and 8 ounces of B grade coffee, and a special blend that contains 7 ounces of A grade coffee and 3 ounces of B grade coffee. Let x be the number of deluxe blend packages sold. Let y be the number of special blend packages sold. Shade the region corresponding to all values of x and y that satisfy these requirements.

Solve Amy’s Coffee House optimization problem using graphing and systems of linear inequalities. Learn how to plot the constraints for A grade and B grade coffee in a graphical representation to find feasible solutions. This problem covers key concepts of linear inequalities and graphing, suitable for students in Grades 9-12.

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