the equations given doesnt have a y-intercept 

You're correct that the equation $ y = 8x $ lacks a non-zero y-intercept, as the intercept is 0 in this case. Let's clarify why that is and what it means in the context of the problem.

### Context of the Problem:
In the given problem, the chef's earnings start at $0 when 0 hours are worked, which means that there is no initial fixed payment—earnings start accumulating only after work begins. This is why the y-intercept is 0.

### Explanation:
- **Y-intercept**: The y-intercept in an equation represents the value of $ y $ when $ x = 0 $. In this case, $ y $ represents earnings, and $ x $ represents hours worked.
  - When $ x = 0 $ (no hours worked), the chef's earnings are also $0, as shown in the table. Therefore, the y-intercept $ b = 0 $.

### Equation Recap:
- The slope $ m = 8 $ indicates the chef earns $8 per hour.
- The y-intercept $ b = 0 $ indicates there is no initial earnings or fee.

Thus, the equation $ y = 8x $ accurately represents the chef's earnings based solely on the hours worked, with no additional starting amount.

### Graphical Interpretation:
If you were to graph this equation, the line would pass through the origin (0, 0), indicating that at 0 hours, the earnings are also 0, and for every additional hour worked, the earnings increase by $8.

Would you like any further clarification or details on this?

### Related Questions:
1. How does the absence of a y-intercept affect the graph of the equation?
2. Can you explain how to calculate the slope if the table had a starting fee (e.g., $20)?
3. If the y-intercept were $10, how would that change the equation?
4. What would the graph of $ y = 8x + 20 $ look like, and how would it differ from $ y = 8x $?
5. Why is the slope constant in this scenario, and what does it signify?
6. How would you interpret the equation if the slope were negative?
7. Could the y-intercept ever be negative in a real-world context? What would that mean?
8. How would the earnings model change if the hourly rate varied after a certain number of hours?

**Tip:** When analyzing linear equations, always consider both the slope and y-intercept to understand the full relationship between the variables involved.

Explore why certain linear equations, like y = 8x, lack a y-intercept. Learn about the concepts of y-intercept, slope, and their graphical interpretations in linear equations. Suitable for Grades 7-9.

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