To solve this problem, we need to set up and solve a system of equations based on the given information.

Let:

• $c$ represent the cost of a cake in dollars.
• $p$ represent the cost of a pie in dollars.

Step 1: Write the system of equations

The two main pieces of information given are:

1. The price of a cake is $9 more than the price of a pie. $c = p + 9$
2. The bakery sold 8 cakes and 14 pies for a total of $402. $8c + 14p = 402$

Step 2: Solve the system of equations

1. From the first equation, we know that $c = p + 9$. We can substitute this into the second equation to eliminate $c$. $8(p + 9) + 14p = 402$ Simplifying this: $8p + 72 + 14p = 402$ Combine like terms: $22p + 72 = 402$ Subtract 72 from both sides: $22p = 330$ Divide by 22: $p = 15$

2. Now that we know $p = 15$, substitute this value into the first equation to find $c$: $c = 15 + 9 = 24$

Step 3: Final Answer

• The cost of a pie is $15.
• The cost of a cake is $24.

Recap of the equations:

1. $c = p + 9$
2. $8c + 14p = 402$

Would you like any further details or have any questions?

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Related Questions:

1. What if the bakery sold more cakes than pies? How would that affect the total price?
2. If the price of the cake increased by 20%, how would that change the system of equations?
3. What would the total revenue be if they sold 10 cakes and 10 pies instead?
4. How can we solve this system of equations using a matrix method?
5. How would you express the total revenue as a function of the number of cakes and pies sold?

Tip: When dealing with word problems involving systems of equations, always define your variables clearly and express relationships between them before solving.