The given situation describes a linear relationship between the amount of money Carrie has in her savings account and the number of weeks. A linear relationship can be presented as a linear equation in the form of y = mx + b, where y represents the dependent variable, m is the slope, x is the independent variable, and b is the y-intercept. In the given situation, 'y' represents the amount of money in Carrie's savings account, 'x' is the number of weeks.
First, let's find the slope of this linear equation using the given two sets of data points: (2, 700) and (5, 325). The slope, or rate of change, in this case, represents the rate at which the amount of money changes each week. It is calculated as the change in y (the account balance) divided by the change in x (time, or number of weeks). Using the formula for the slope of a line:
m = (y2 - y1) / (x2 - x1)
where
x1 = 2 weeks, y1 = $700 (the amount of money in the second week),
x2 = 5 weeks, y2 = $325 (the amount of money in the fifth week).
Substituting the given values into the formula, we get:
m = (325 - 700) / (5 - 2) = -125
This means that the amount of money in Carrie's account decreases by $125 each week.
Next, we calculate the y-intercept, which is the value of 'y' when 'x' equals zero. This represents the initial account balance in the first week of the year. We can use the formula:
b = y1 - m * x1
Substituting the known values gives us:
b = 700 - (-125) * 2 = 950
Therefore, the initial amount of money in her savings account was $950.
Now that we have found the slope 'm' and the y-intercept 'b', we can write down the equation that represents this situation:
y = -125x + 950
In this equation, 'y' is the amount of money in Carrie's savings account at the end of x weeks, and 'x' is the number of weeks since the start of the year.