Question

emery borrowed money from her brother to buy a new phone and is paying off a fixed amount each week. after 2 weeks, she will owe $456, and after 5 weeks, she will owe $228.
a. what was the original amount emery borrowed?
b. how much does she pay each week?
c. how useful are equations in point - slope and slope - intercept forms for answering each question?
a. emery originally borrowed $

Explanation:

Step1: Find the payment per - week (slope)

The change in the amount owed is $\Delta y=456 - 228=228$, and the change in the number of weeks is $\Delta x=5 - 2 = 3$. The amount paid per week (slope $m$) is given by the formula $m=\frac{\Delta y}{\Delta x}$. So, $m=\frac{456 - 228}{5 - 2}=\frac{228}{3}=76$.

Step2: Use the point - slope form to find the original amount

The point - slope form of a line is $y - y_1=m(x - x_1)$. Let $x$ be the number of weeks and $y$ be the amount owed. We can use the point $(2,456)$. The equation of the line is $y-456 = 76(x - 2)$. To find the original amount (when $x = 0$), we substitute $x = 0$ into the equation: $y-456=76(0 - 2)$, $y-456=-152$, $y=456 + 152=608$.

Step3: Analyze the usefulness of the forms

The point - slope form $y - y_1=m(x - x_1)$ is useful for finding the original amount. We can choose a known point $(x_1,y_1)$ (like $(2,456)$ or $(5,228)$) and the slope $m$ to write the equation of the line representing the amount owed over time. The slope - intercept form $y=mx + b$ is also useful as once we find the slope $m$ and the $y$ - intercept $b$ (the original amount), we can easily answer questions about the amount owed at different times.

Answer:

a. $608$
b. $76$
c. The point - slope form and slope - intercept form are very useful. The point - slope form helps in writing the equation of the line using a known point and slope, and the slope - intercept form gives a clear relationship between the number of weeks and the amount owed, with the $y$ - intercept being the original amount borrowed.