Final answer:
The linear function that models the bank account balance over time is y = 200x + 0, where the slope is 200, indicating the account increases by $200 each week. The y-intercept is 0, which represents the starting balance of the account.
Step-by-step explanation:
To write an equation for the line that models the bank account's balance over time with a constant rate of change, we can use two points that we know: (2, 400) and (5, 1000). The first number represents the number of weeks after the account was opened, and the second number represents the balance of the account at that time.
The slope of the line can be found using the formula m = (y2 - y1) / (x2 - x1), which in this case is (1000 - 400) / (5 - 2) = 600 / 3 = 200. This slope means the bank account balance increases by $200 each week.
The y-intercept is the balance of the account when x (the number of weeks) is 0. To find the y-intercept, we can use the slope-intercept form of a linear function, y = mx + b, where m is the slope and b is the y-intercept. By substituting one of the points and the slope into the equation, we get 400 = 200(2) + b, which simplifies to b = $0. However, the y-intercept doesn't make sense in the context as the account could not have started with $0 and reached $400 after 2 weeks. Instead, we can use the point-slope form, y - y1 = m(x - x1), using the point (2,400) and simplify it to get the equation in slope-intercept form.
Therefore, the equation of the line is y = 200x + 0. The slope of the equation represents the weekly increase in the bank account balance, and the y-intercept represents the balance when the account was first opened.