The problem involves setting up and solving a system of equations to determine the quantity of each product (apple pie and pumpkin pie) for Isabella and William based on the given total servings and total price.

Isabella's System of Equations

1. Total servings equation: $x + y = 15$ Where:

   • $x$: servings of apple pie
   • $y$: servings of pumpkin pie

2. Total price equation: $3.7x + 3.3y = 56.20$ Here, $3.70 is the price per serving of apple pie, and $3.30 is the price per serving of pumpkin pie.

William's System of Equations

1. Total servings equation: $u + w = 15$ Where:

   • $u$: servings of apple pie
   • $w$: servings of pumpkin pie

2. Total price equation: $3.7u + 3.3w = 56.20$

Solving the Systems

The systems for Isabella and William are structured similarly but will be solved separately. Let’s solve Isabella’s equations as an example:

Step 1: Solve one equation for one variable

From the total servings equation ($x + y = 15$): $y = 15 - x$

Step 2: Substitute into the total price equation

Substitute $y = 15 - x$ into $3.7x + 3.3y = 56.20$: $3.7x + 3.3(15 - x) = 56.20$

Step 3: Simplify and solve for $x$

$3.7x + 49.5 - 3.3x = 56.20$ $0.4x + 49.5 = 56.20$ $0.4x = 6.70$ $x = \frac{6.70}{0.4} = 16.75$

Step 4: Solve for $y$

Using $y = 15 - x$: $y = 15 - 16.75 = -1.75$

This result suggests some inconsistencies in the given problem setup or the prices, as the servings cannot be negative. Would you like to proceed further in checking both systems for inconsistencies or reviewing the problem details further?

Feel free to ask!

Further Questions You Might Explore:

1. How do we interpret solutions with inconsistencies?
2. Can we adjust the prices or servings to create a feasible solution?
3. Should we try solving William's equations to see if they align?
4. Could the total price be distributed differently between the pie types?
5. What assumptions might we adjust to make the system realistic?

Tip:

When solving systems of equations with decimals, consider multiplying through by 10 or 100 to simplify calculations!