Question
3) The cost of a gym membership can be represented by 25 + 5x, where x is the number of visits to the gym.
How many times can you visit if you have a $100 budget?
What is the y-intercept? What does it mean in context?
What is the slope? What does it mean in context?
How many times can you visit if you have a $100 budget?
What is the y-intercept? What does it mean in context?
What is the slope? What does it mean in context?
Answers
GPT-4o mini
To find out how many times you can visit the gym within a $100 budget using the cost equation \( C = 25 + 5x \), we need to set the equation equal to 100 and solve for \( x \).
1. **Set up the equation:**
\[
100 = 25 + 5x
\]
2. **Subtract 25 from both sides:**
\[
100 - 25 = 5x
\]
\[
75 = 5x
\]
3. **Divide both sides by 5 to solve for \( x \):**
\[
x = \frac{75}{5} = 15
\]
So, you can visit the gym **15 times** within a $100 budget.
---
### Y-Intercept
The y-intercept occurs when \( x = 0 \). In the equation \( C = 25 + 5x \), if we set \( x = 0 \):
\[
C = 25 + 5(0) = 25
\]
- **Y-intercept**: \( C = 25 \)
**Meaning in context**: The y-intercept of 25 represents the fixed cost of the gym membership, which is the amount you need to pay to have access to the gym, even if you do not visit it at all. This is a cost associated with being a member of the gym.
---
### Slope
The slope of the equation \( C = 25 + 5x \) is the coefficient of \( x \), which is \( 5 \).
- **Slope**: 5
**Meaning in context**: The slope indicates that for every additional visit to the gym (every increase of 1 in \( x \)), the total cost increases by $5. Therefore, each visit to the gym incurs an additional charge of $5 on top of the fixed membership fee.
1. **Set up the equation:**
\[
100 = 25 + 5x
\]
2. **Subtract 25 from both sides:**
\[
100 - 25 = 5x
\]
\[
75 = 5x
\]
3. **Divide both sides by 5 to solve for \( x \):**
\[
x = \frac{75}{5} = 15
\]
So, you can visit the gym **15 times** within a $100 budget.
---
### Y-Intercept
The y-intercept occurs when \( x = 0 \). In the equation \( C = 25 + 5x \), if we set \( x = 0 \):
\[
C = 25 + 5(0) = 25
\]
- **Y-intercept**: \( C = 25 \)
**Meaning in context**: The y-intercept of 25 represents the fixed cost of the gym membership, which is the amount you need to pay to have access to the gym, even if you do not visit it at all. This is a cost associated with being a member of the gym.
---
### Slope
The slope of the equation \( C = 25 + 5x \) is the coefficient of \( x \), which is \( 5 \).
- **Slope**: 5
**Meaning in context**: The slope indicates that for every additional visit to the gym (every increase of 1 in \( x \)), the total cost increases by $5. Therefore, each visit to the gym incurs an additional charge of $5 on top of the fixed membership fee.