Interpretation of the Image:

The provided image outlines Activity 1: Five Savings Accounts, which includes a graph showing several linear relationships (labeled as A, B, C, D, and H). Each line represents a person's weekly savings account balance over the first 17 weeks of the year. The slope and intercepts of these lines correspond to the savings rate and starting balance.

Tasks Identified in the Activity:

1. Part a: Select one line and describe its progression.
2. Part b: Share your description to identify the line.
3. Part c: Write an equation for each line and interpret the slope ($m$) and y-intercept ($b$).

Part a:

You selected line H, which increases with a slope ($m$) of 20 and starts at $-80$ (y-intercept). This indicates that the savings balance began at $-80$ dollars and grew by 20 dollars each week.

Part c: Writing Equations

General Form:

The equation for a line is given as: $y = mx + b$ Where:

• $m$ = slope (rate of change of savings),
• $b$ = y-intercept (starting savings balance),
• $y$ = total savings at week $x$.

Deriving Equations:

From the graph, let's extract information for each line:

1. Line H:

   • Slope ($m$): 20 (increase of 20 per week),
   • y-Intercept ($b$): $-80$,
   • Equation: $y = 20x - 80$.

2. Line A:

   • Slope ($m$): 10,
   • y-Intercept ($b$): 20,
   • Equation: $y = 10x + 20$.

3. Line B:

   • Slope ($m$): 0 (balance remains constant),
   • y-Intercept ($b$): 80,
   • Equation: $y = 80$.

4. Line C:

   • Slope ($m$): $-20$ (decrease of 20 per week),
   • y-Intercept ($b$): 120,
   • Equation: $y = -20x + 120$.

5. Line D:

   • Slope ($m$): $-10$ (decrease of 10 per week),
   • y-Intercept ($b$): 100,
   • Equation: $y = -10x + 100$.

Interpretations of Slope and Intercept:

• Slope ($m$): Represents the weekly change in the savings account balance (positive for increasing, negative for decreasing, zero for constant balance).
• Y-Intercept ($b$): The starting balance at week 0.

Question d:

• For which equation is $y = 70$ a solution?
  • Plug $y = 70$ into each equation to find corresponding $x$ values:
    • For $y = 20x - 80$, solve $70 = 20x - 80$: $x = 7.5$.
    • For $y = 10x + 20$, solve $70 = 10x + 20$: $x = 5$.
    • For $y = 80$, no solution (balance is constant at 80).
    • For $y = -20x + 120$, solve $70 = -20x + 120$: $x = 2.5$.
    • For $y = -10x + 100$, solve $70 = -10x + 100$: $x = 3$.

Do you need further clarifications or detailed explanations?

Here are 5 related questions to deepen understanding:

1. How would a slope of $0$ (e.g., line B) differ in interpretation compared to positive or negative slopes?
2. Can you calculate when each line reaches a specific balance, such as $y = 0$?
3. How does the y-intercept influence the line's position on the graph?
4. What happens to a line's equation if the savings rate (slope) doubles?
5. How can these equations predict savings beyond the 17th week?

Tip: To confirm your equations, test them by plugging in points (x, y) from the graph!