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. 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?
Answers
GPT-4o mini
To solve this problem, let's start by defining a few variables:
- Let \( P \) be the original amount Emery borrowed.
- Let \( w \) be the fixed amount Emery pays back each week.
- Let \( t \) be the number of weeks that have passed.
According to the information provided:
1. After 2 weeks, Emery owes $456. This can be expressed as:
\[
P - 2w = 456 \quad \text{(1)}
\]
2. After 5 weeks, she owes $228, which can be expressed as:
\[
P - 5w = 228 \quad \text{(2)}
\]
Now we can solve these equations step by step.
### Part a: Finding the Original Amount Borrowed (P)
We have the two equations:
1. \( P - 2w = 456 \) (equation 1)
2. \( P - 5w = 228 \) (equation 2)
To eliminate \( P \), we can subtract equation (2) from equation (1):
\[
(P - 2w) - (P - 5w) = 456 - 228
\]
This simplifies to:
\[
-2w + 5w = 456 - 228
\]
\[
3w = 228
\]
Now, solve for \( w \):
\[
w = \frac{228}{3} = 76
\]
Now that we know \( w \), we can substitute \( w = 76 \) back into either equation to find \( P \). We'll use equation (1):
\[
P - 2(76) = 456
\]
\[
P - 152 = 456
\]
\[
P = 456 + 152
\]
\[
P = 608
\]
Thus, the original amount Emery borrowed is:
\[
\boxed{608}
\]
### Part b: Finding the Fixed Amount Paid Each Week (w)
From the previous calculations, we found that her weekly payment is:
\[
\boxed{76}
\]
### Part c: Usefulness of Equations in Point-Slope and Slope-Intercept Forms
**Point-slope form** of a line is generally expressed as:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope of the line and \( (x_1, y_1) \) is a known point on the line. In this problem, we could use the amounts owed after 2 weeks (week = 2, amount owed = 456) and after 5 weeks (week = 5, amount owed = 228) to create equations and find the relationship between the weeks and amount owed.
**Slope-intercept form** is expressed as:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept. This form is useful for determining how the amounts owed change over time (with respect to the number of weeks) and for predicting future amounts owed.
Both forms are useful for this problem; they can provide insight into how the debt decreases over time by showing the relationship between time (weeks) and the amount remaining owed.
The linear equations created from the problem perfectly fit the linear relationship depicted in both forms, showcasing how useful equations can be for understanding and calculating real-life financial scenarios like this one.
- Let \( P \) be the original amount Emery borrowed.
- Let \( w \) be the fixed amount Emery pays back each week.
- Let \( t \) be the number of weeks that have passed.
According to the information provided:
1. After 2 weeks, Emery owes $456. This can be expressed as:
\[
P - 2w = 456 \quad \text{(1)}
\]
2. After 5 weeks, she owes $228, which can be expressed as:
\[
P - 5w = 228 \quad \text{(2)}
\]
Now we can solve these equations step by step.
### Part a: Finding the Original Amount Borrowed (P)
We have the two equations:
1. \( P - 2w = 456 \) (equation 1)
2. \( P - 5w = 228 \) (equation 2)
To eliminate \( P \), we can subtract equation (2) from equation (1):
\[
(P - 2w) - (P - 5w) = 456 - 228
\]
This simplifies to:
\[
-2w + 5w = 456 - 228
\]
\[
3w = 228
\]
Now, solve for \( w \):
\[
w = \frac{228}{3} = 76
\]
Now that we know \( w \), we can substitute \( w = 76 \) back into either equation to find \( P \). We'll use equation (1):
\[
P - 2(76) = 456
\]
\[
P - 152 = 456
\]
\[
P = 456 + 152
\]
\[
P = 608
\]
Thus, the original amount Emery borrowed is:
\[
\boxed{608}
\]
### Part b: Finding the Fixed Amount Paid Each Week (w)
From the previous calculations, we found that her weekly payment is:
\[
\boxed{76}
\]
### Part c: Usefulness of Equations in Point-Slope and Slope-Intercept Forms
**Point-slope form** of a line is generally expressed as:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope of the line and \( (x_1, y_1) \) is a known point on the line. In this problem, we could use the amounts owed after 2 weeks (week = 2, amount owed = 456) and after 5 weeks (week = 5, amount owed = 228) to create equations and find the relationship between the weeks and amount owed.
**Slope-intercept form** is expressed as:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept. This form is useful for determining how the amounts owed change over time (with respect to the number of weeks) and for predicting future amounts owed.
Both forms are useful for this problem; they can provide insight into how the debt decreases over time by showing the relationship between time (weeks) and the amount remaining owed.
The linear equations created from the problem perfectly fit the linear relationship depicted in both forms, showcasing how useful equations can be for understanding and calculating real-life financial scenarios like this one.