Question
Elijah is saving money to buy a new gaming system.He set aside some money he received for his birthday and then added more each week. The table shows the amount of money Elijah has saved at the end of each week, m, for the number of week he has been saving, w.
the table:
w 1, 2, 5, 7
m, 80, 110, 200, 260
use the information to determine the slope and y -intercept in this relationship. The. interpret each value in context
the table:
w 1, 2, 5, 7
m, 80, 110, 200, 260
use the information to determine the slope and y -intercept in this relationship. The. interpret each value in context
Answers
GPT-4o mini
To determine the slope and y-intercept of the relationship between the number of weeks Elijah has been saving (w) and the amount of money saved (m), we can analyze the values from the table. We can assume the relationship is linear, which means we can express it in the form of a linear equation:
\[ m = mw + b \]
where:
- \( m \) is the amount saved,
- \( w \) is the number of weeks,
- \( m \) is the slope (rate of change of amount saved per week), and
- \( b \) is the y-intercept (the initial amount of money saved).
### Step 1: Determine the slope (m)
To calculate the slope, we can select two points from the table. Let's use the points (1, 80) and (2, 110).
The formula for the slope (m) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points:
- \( (1, 80) \) → \( (w_1 = 1, m_1 = 80) \)
- \( (2, 110) \) → \( (w_2 = 2, m_2 = 110) \)
Substituting these values into the slope formula:
\[
m = \frac{110 - 80}{2 - 1} = \frac{30}{1} = 30
\]
So, the slope \( m = 30 \).
### Step 2: Determine the y-intercept (b)
To find the y-intercept, we can use one of the points in the form of the equation we established earlier. The slope-intercept equation can be used as \( m = 30w + b \). We'll use the point \( (1, 80) \):
\[
80 = 30(1) + b
\]
Solving for \( b \):
\[
80 = 30 + b \\
b = 80 - 30 \\
b = 50
\]
### Summary of Results
1. **Slope (m): 30**
- This means Elijah saves $30 each week after initially setting aside money. It represents the rate at which Elijah's savings increase as he continues to save weekly.
2. **Y-intercept (b): 50**
- This represents the initial amount of money Elijah had set aside before he began saving weekly. This means that at week 0 (the starting point), he had $50.
### Final Interpretation
In summary:
- Elijah started with **$50** before saving began, which is reflected in the y-intercept.
- With each passing week, he adds **$30** to his savings, which is shown by the slope of the line.
\[ m = mw + b \]
where:
- \( m \) is the amount saved,
- \( w \) is the number of weeks,
- \( m \) is the slope (rate of change of amount saved per week), and
- \( b \) is the y-intercept (the initial amount of money saved).
### Step 1: Determine the slope (m)
To calculate the slope, we can select two points from the table. Let's use the points (1, 80) and (2, 110).
The formula for the slope (m) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points:
- \( (1, 80) \) → \( (w_1 = 1, m_1 = 80) \)
- \( (2, 110) \) → \( (w_2 = 2, m_2 = 110) \)
Substituting these values into the slope formula:
\[
m = \frac{110 - 80}{2 - 1} = \frac{30}{1} = 30
\]
So, the slope \( m = 30 \).
### Step 2: Determine the y-intercept (b)
To find the y-intercept, we can use one of the points in the form of the equation we established earlier. The slope-intercept equation can be used as \( m = 30w + b \). We'll use the point \( (1, 80) \):
\[
80 = 30(1) + b
\]
Solving for \( b \):
\[
80 = 30 + b \\
b = 80 - 30 \\
b = 50
\]
### Summary of Results
1. **Slope (m): 30**
- This means Elijah saves $30 each week after initially setting aside money. It represents the rate at which Elijah's savings increase as he continues to save weekly.
2. **Y-intercept (b): 50**
- This represents the initial amount of money Elijah had set aside before he began saving weekly. This means that at week 0 (the starting point), he had $50.
### Final Interpretation
In summary:
- Elijah started with **$50** before saving began, which is reflected in the y-intercept.
- With each passing week, he adds **$30** to his savings, which is shown by the slope of the line.