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The problem discusses Julio's costs and revenue for hosting a pottery painting party. Two linear equations are given:

1. Julio's **cost** equation:  
   \[ y = 100 + 7x \]  
   This represents the cost of the party, where it costs $100 plus $7 per guest.  
   - $ x $ is the number of guests.
   - $ y $ is the total cost in dollars.

2. Julio's **revenue** equation:  
   \[ y = 50 + 12x \]  
   This represents the money Julio charges for the party, which includes a $50 fee and $12 per guest.

The goal is to find out how many guests Julio needs to host to **break even**, meaning when his costs equal his revenue.

### Step 1: Set the two equations equal to each other to solve for $ x $ (the number of guests):
\[
100 + 7x = 50 + 12x
\]

### Step 2: Solve for $ x $:
\[
100 - 50 = 12x - 7x
\]
\[
50 = 5x
\]
\[
x = \frac{50}{5} = 10
\]

### Conclusion:
Jul

Julio owns a pottery painting studio and offers parties where guests can paint their own pieces of pottery to take home. For one party, it costs Julio $100 plus an additional $7 per person. He charges party hosts an initial fee of $50 plus $12 for each person who attends. How many guests must Julio host to break even?

Learn how to solve a system of linear equations involving costs and revenue for Julio's pottery painting party. This step-by-step solution explains how to determine the number of guests needed for Julio to break even. Suitable for students in Grades 8-10.

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