Mario was given some birthday money. He puts the money in an account. Every month after that, he deposits the same amount of money.

The equation that models this situation is [tex]$y = 75x + 50[tex]$[/tex], where [tex]$[/tex]y[tex]$[/tex] is the amount of money in the account and [tex]$[/tex]x$[/tex] is the number of deposits.

What does the [tex]$y$[/tex]-intercept mean in this situation?

A. He puts [tex]$\$[/tex]50$ in the account each month.
B. He was given [tex]$\$[/tex]75$ for his birthday.
C. He was given [tex]$\$[/tex]50$ for his birthday.
D. He puts [tex]$\$[/tex]75$ in the account each month.



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.