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.

Question

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?

Answer 1

To solve the problem, we can start by letting the original amount Emery borrowed be \( P \) and the fixed amount she pays each week be \( d \).

After 2 weeks, the amount she owes, which is \( P - 2d \), is given to be $456:

1. \( P - 2d = 456 \) (Equation 1)

After 5 weeks, the amount she owes, which is \( P - 5d \), is given to be $228:

2. \( P - 5d = 228 \) (Equation 2)

a. What was the original amount Emery borrowed?

We can solve this system of equations (Equation 1 and Equation 2).

First, let's rearrange both equations to isolate \( P \):

From Equation 1: \[ P = 456 + 2d \quad (Equation 3) \]

From Equation 2: \[ P = 228 + 5d \quad (Equation 4) \]

Now, we can set Equation 3 equal to Equation 4:

\[ 456 + 2d = 228 + 5d \]

Now, solve for \( d \):

\[ 456 - 228 = 5d - 2d \] \[ 228 = 3d \] \[ d = \frac{228}{3} = 76 \]

Now that we have \( d \), we can substitute this value back into either Equation 3 or Equation 4 to find \( P \). Using Equation 3:

\[ P = 456 + 2(76) \] \[ P = 456 + 152 = 608 \]

So, the original amount Emery borrowed is $608.

b. How much does she pay each week?

From our calculation, the fixed amount she pays each week is $76.

c. How useful are equations in point-slope and slope-intercept forms for answering each question?

Equations in point-slope and slope-intercept forms can be useful in this context as they help visualize the relationship between the amount owed and the number of weeks. Here's how they can relate:

1. Slope-Intercept Form (y = mx + b): You could express the amount owed \( y \) as a function of weeks \( x \) with a slope \( m \) representing the amount paid per week (which we found to be $76), and \( b \) represents the original amount borrowed. Knowing that the amount decreases linearly as she pays off the loan makes this form useful for visual understanding of how the debt decreases over time.

2. Point-Slope Form (y - y_1 = m(x - x_1)): This form can be useful if you have a specific point (for example, after 2 weeks or 5 weeks) and the slope (weekly payment). For instance, you could use the point where \( (2, 456) \) and the slope \( -76 \) to describe how the amount owed changes week by week.

In conclusion, both forms provide a clear and systematic way to analyze the relationships involved, allowing for easier understanding and prediction of the amount owed over time.