Answer :
To understand what the \( y \)-intercept represents in this context, let’s carefully analyze the given equation:
[tex]\[ y = 75x + 50 \][/tex]
In this equation:
- \( y \) represents the total amount of money in Mario's account.
- \( x \) represents the number of monthly deposits.
- The coefficient \( 75 \) is the amount of money Mario deposits each month.
- The constant \( 50 \) is the \( y \)-intercept of the equation.
### Understanding the \( y \)-Intercept:
1. Definition: The \( y \)-intercept in a linear equation \( y = mx + b \) is the point where the line crosses the \( y \)-axis. This happens when \( x = 0 \).
2. Interpretation in the Context: When \( x = 0 \), this means that no monthly deposits have been made yet. Therefore, the \( y \)-intercept represents the initial amount of money in the account before any additional deposits.
### Applying to the Given Situation:
- According to the equation \( y = 75x + 50 \), when \( x = 0 \):
[tex]\[ y = 75 \cdot 0 + 50 \][/tex]
[tex]\[ y = 50 \][/tex]
- This initial amount of \( \$50 \) is the money Mario had in his account before making any monthly deposits.
### Conclusion:
From these observations, we can conclude that the \( y \)-intercept (which is \( 50 \)) represents the initial amount of money Mario received for his birthday, before any monthly deposits were made.
### Answer:
The correct interpretation is:
C. He was given \$50 for his birthday.
This means the [tex]\( \$50 \)[/tex] accounts for the birthday money Mario started with in his account.
[tex]\[ y = 75x + 50 \][/tex]
In this equation:
- \( y \) represents the total amount of money in Mario's account.
- \( x \) represents the number of monthly deposits.
- The coefficient \( 75 \) is the amount of money Mario deposits each month.
- The constant \( 50 \) is the \( y \)-intercept of the equation.
### Understanding the \( y \)-Intercept:
1. Definition: The \( y \)-intercept in a linear equation \( y = mx + b \) is the point where the line crosses the \( y \)-axis. This happens when \( x = 0 \).
2. Interpretation in the Context: When \( x = 0 \), this means that no monthly deposits have been made yet. Therefore, the \( y \)-intercept represents the initial amount of money in the account before any additional deposits.
### Applying to the Given Situation:
- According to the equation \( y = 75x + 50 \), when \( x = 0 \):
[tex]\[ y = 75 \cdot 0 + 50 \][/tex]
[tex]\[ y = 50 \][/tex]
- This initial amount of \( \$50 \) is the money Mario had in his account before making any monthly deposits.
### Conclusion:
From these observations, we can conclude that the \( y \)-intercept (which is \( 50 \)) represents the initial amount of money Mario received for his birthday, before any monthly deposits were made.
### Answer:
The correct interpretation is:
C. He was given \$50 for his birthday.
This means the [tex]\( \$50 \)[/tex] accounts for the birthday money Mario started with in his account.
