Question
The table below shows the amount that must be repaid, y, when x dollars are borrowed.
Amount Borrowed Amount Repaid
$100 $135
$200 $260
$500 $635
$800 $1,010
(1 point)
Responses
y=54x
y is equal to 5 fourths x
y=45x+55
y is equal to 4 fifths x plus 55
y=54x+10
y is equal to 5 fourths x plus 10
y=45x
Amount Borrowed Amount Repaid
$100 $135
$200 $260
$500 $635
$800 $1,010
(1 point)
Responses
y=54x
y is equal to 5 fourths x
y=45x+55
y is equal to 4 fifths x plus 55
y=54x+10
y is equal to 5 fourths x plus 10
y=45x
Answers
GPT-4o mini
To determine the correct relationship between the amount borrowed (x) and the amount repaid (y), we can look for a pattern in the table provided:
1. **Calculating the Repayment Rate or Equation**:
Let's first calculate the rate of repayment for each amount borrowed:
- For $100 borrowed, $135 is repaid:
\[
\text{Rate} = \frac{y}{x} = \frac{135}{100} = 1.35
\]
- For $200 borrowed, $260 is repaid:
\[
\text{Rate} = \frac{260}{200} = 1.3
\]
- For $500 borrowed, $635 is repaid:
\[
\text{Rate} = \frac{635}{500} = 1.27
\]
- For $800 borrowed, $1,010 is repaid:
\[
\text{Rate} = \frac{1010}{800} = 1.2625
\]
2. **Finding the Linear Equation**: To derive a linear equation \(y = mx + b\), let's look for a linear pattern in the derived points.
We can calculate the differences:
- The differences in amount borrowed and amount repaid:
- From $100 to $200, the increase is $65.
- From $200 to $500, the increase is $375.
- From $500 to $800, the increase is $375.
This shows the relationships are not purely linear based on these increments. Therefore, let's calculate the overall relationship assuming a linear equation.
3. **Finding the Slope (m) and Y-intercept (b)**:
Using two points to find the slope:
The two points are: (100, 135) and (500, 635).
- The slope \(m\) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{635 - 135}{500 - 100} = \frac{500}{400} = \frac{5}{4} = 1.25
\]
So, we can say:
\[
y = \frac{5}{4}x + b
\]
To find \(b\), we can substitute one of the points in:
Choosing the point (100, 135):
\[
135 = \frac{5}{4}(100) + b \implies 135 = 125 + b \implies b = 135 - 125 = 10.
\]
Thus, the equation becomes:
\[
y = \frac{5}{4}x + 10
\]
4. **Final Conclusion**:
The closest matching equation from your choices is **y = 54x + 10**, which suggests you may be asked to represent the slope differently \(y = \frac{5}{4}x + 10\).
Considering \(54x\) directly does not appear which suggests a linear relationship of amount borrowed with an additional flat fee of $10, confirming we reviewed each of the options.
Thus the correct equation should be:
\[
\boxed{y = \frac{5}{4}x + 10}
\]
1. **Calculating the Repayment Rate or Equation**:
Let's first calculate the rate of repayment for each amount borrowed:
- For $100 borrowed, $135 is repaid:
\[
\text{Rate} = \frac{y}{x} = \frac{135}{100} = 1.35
\]
- For $200 borrowed, $260 is repaid:
\[
\text{Rate} = \frac{260}{200} = 1.3
\]
- For $500 borrowed, $635 is repaid:
\[
\text{Rate} = \frac{635}{500} = 1.27
\]
- For $800 borrowed, $1,010 is repaid:
\[
\text{Rate} = \frac{1010}{800} = 1.2625
\]
2. **Finding the Linear Equation**: To derive a linear equation \(y = mx + b\), let's look for a linear pattern in the derived points.
We can calculate the differences:
- The differences in amount borrowed and amount repaid:
- From $100 to $200, the increase is $65.
- From $200 to $500, the increase is $375.
- From $500 to $800, the increase is $375.
This shows the relationships are not purely linear based on these increments. Therefore, let's calculate the overall relationship assuming a linear equation.
3. **Finding the Slope (m) and Y-intercept (b)**:
Using two points to find the slope:
The two points are: (100, 135) and (500, 635).
- The slope \(m\) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{635 - 135}{500 - 100} = \frac{500}{400} = \frac{5}{4} = 1.25
\]
So, we can say:
\[
y = \frac{5}{4}x + b
\]
To find \(b\), we can substitute one of the points in:
Choosing the point (100, 135):
\[
135 = \frac{5}{4}(100) + b \implies 135 = 125 + b \implies b = 135 - 125 = 10.
\]
Thus, the equation becomes:
\[
y = \frac{5}{4}x + 10
\]
4. **Final Conclusion**:
The closest matching equation from your choices is **y = 54x + 10**, which suggests you may be asked to represent the slope differently \(y = \frac{5}{4}x + 10\).
Considering \(54x\) directly does not appear which suggests a linear relationship of amount borrowed with an additional flat fee of $10, confirming we reviewed each of the options.
Thus the correct equation should be:
\[
\boxed{y = \frac{5}{4}x + 10}
\]