In the equation $y = -2x + 100$, which could represent the balance in a savings account after spending a fixed amount each week, the slope and y-intercept have specific interpretations in this context.

Slope (-2)

The slope, $-2$, represents the rate of change in the balance of the savings account per week. Since it is negative, it indicates a decrease in the balance over time. Specifically, this slope tells us that every week, the balance decreases by 2 units. If we assume "units" to be dollars, this would mean that $2 is spent from the account each week.

In other words, for each additional week (as $x$ increases by 1), the balance $y$ decreases by 2 dollars.

Y-intercept (100)

The y-intercept, $100$, represents the starting balance in the savings account when $x = 0$, which is the initial time or the beginning of the observation period. In this context, the y-intercept indicates that the account started with a balance of $100.

Interpretation Summary

If this equation models the balance in a savings account where a fixed amount is spent each week:

• The slope of $-2$ implies a weekly expenditure of $2, leading to a steady decline in the balance.
• The y-intercept of $100$ indicates that the initial balance in the savings account was $100 at the beginning of this period.

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Related Questions:

1. How would the equation change if $3 were spent each week instead?
2. What would the slope be if the account balance increased by $5 each week instead of decreasing?
3. How would the balance change after 10 weeks, according to this equation?
4. What would the equation look like if the initial balance were $200 instead of $100?
5. If the goal is to keep the account balance above zero, how many weeks would it take for the balance to reach zero?

Tip:

When analyzing linear equations in real-world contexts, the slope often represents a rate (like spending or earning per time period), and the y-intercept represents the initial amount or starting point.