A party planner can spend a maximum of $5000 on food. If the chicken dinner (x) costs $20 and the steak dinner (y) costs $25, make a graph of the region that shows the possibilities for the number of chicken and steak dinners that can be purchased while still staying within budget. Shade the solution set.

20x+25y <= 5000

type that in at wolframalpha.com and it will show the shaded region

To graph the region that shows the possibilities for the number of chicken and steak dinners that can be purchased while still staying within the budget, we need to define the constraints.

Let's assume we can purchase x chicken dinners and y steak dinners. The cost of one chicken dinner is $20, and the cost of one steak dinner is $25. The total budget available is $5000.

Now, let's formulate the constraints mathematically:

1. Cost constraint: The total cost of the dinners should not exceed the budget of $5000.
Constraint equation: 20x + 25y ≤ 5000

2. Non-negativity constraint: The number of dinners cannot be negative.
Constraint equations: x ≥ 0, y ≥ 0

To graph this region, we need to plot the line representing the cost constraint equation, as well as shade the area of the graph that satisfies the constraints.

First, let's rearrange the cost constraint equation to solve for y:
25y ≤ 5000 - 20x
y ≤ (5000 - 20x)/25
y ≤ (200 - 0.8x)

Now, let's plot this equation on a coordinate system. We'll assume x values from 0 to 250.

Next, we need to shade the region that satisfies all the constraints. This includes the area below the line we just plotted, as well as the area where x and y are both greater than or equal to 0.

The resulting graph will vary depending on the scale used for x and y, but it should be a region bounded by the x-axis, y-axis, and the line we plotted.

Note: The graph may not be a perfect rectangle due to the constraint equations, which only allow non-negative values for x and y.

I hope this explanation helps in creating the graph and understanding the solution set.