In this problem, we need to derive a linear equation that models the number of marbles Marcey has after a given number of weeks. The table lists the number of marbles after specific weeks, which allows us to determine the initial number of marbles (y-intercept) and the weekly rate of increase in marbles (slope).

Step-by-Step Solution:

1. Determine the Y-intercept: The y-intercept represents the initial number of marbles Marcey started with before any weeks passed. Looking at the table, this is the number of marbles when $x = 0$. However, since there's no entry for $x = 0$, we estimate it from the given data by checking values near the beginning, or it might be given explicitly if the problem states it as such.

   Based on the options provided in the image, we are trying to correct an incorrect value of $y$-intercept that was initially assumed as 97. From the data, it appears the corrected value might be 67 based on calculations done for values near $x = 1$.

2. Calculate the Slope: The slope represents the rate of change in the number of marbles per week. Using two points from the table:

   • Point 1: $(1, 77)$
   • Point 2: $(3, 97)$

   The formula for slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the values: $m = \frac{97 - 77}{3 - 1} = \frac{20}{2} = 10$ So, the slope $m = 10$, which represents the number of marbles added each week.

3. Formulate the Equation: Using the slope-intercept form $y = mx + b$, where:

   • $m = 10$
   • $b = 67$ (if 67 is confirmed as the y-intercept)

   The equation becomes: $y = 10x + 67$

4. Interpretation:

   • Slope (10): This represents the number of marbles added each week.
   • Y-intercept (67): This represents the initial number of marbles Marcey had at the start.

Final Equation:

$y = 10x + 67$

Would you like further details on any of these steps, or have additional questions?

Related Questions to Explore:

1. How would the equation change if Marcey started with a different initial number of marbles?
2. What happens to the slope if Marcey adds a different number of marbles each week?
3. How can we verify if a point in the table fits this linear equation?
4. How would you plot this equation on a graph to visually confirm it?
5. If Marcey stops adding marbles after a certain week, how would that change the equation?

Tip:

In any linear equation $y = mx + b$, the slope $m$ always represents the rate of change, and $b$ represents the starting value or initial condition.