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.