Final answer:
The equation 5x + 20y = 500 represents the relationship between the number of bath packages (x) and full grooming packages (y) needed to achieve a total profit of $500. There are multiple solutions to this equation, representing different combinations of both packages that Pat can sell to reach the desired profit.
Step-by-step explanation:
To determine the number of bath packages and full grooming packages that Pat must sell to achieve a total profit of $500, we can set up a system of equations based on the given profit amounts per package. Let x represent the number of bath packages and y represent the number of full grooming packages. Since the profit for each bath package is $5 and for each full grooming package is $20, and Pat wants to make a total profit of $500, our equation will be:
5x + 20y = 500
This equation does not have a unique solution since it represents a linear relationship between x and y. There are multiple combinations of bath packages and full grooming packages that satisfy this equation. Each value of x will determine a corresponding value of y that meets Pat's profit goal. As an example, if Pat sells 100 bath packages (x=100), no full grooming packages are needed (y=0). Alternatively, if 25 full grooming packages are sold (y=25), no bath packages are necessary (x=0). Other combinations of x and y can also be calculated, such as 50 bath packages and 15 full grooming packages.