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], where [tex]y[/tex] is the amount of money in the account and [tex]x[/tex] is the number of deposits.

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

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



Answer :

To understand what the [tex]\( y \)[/tex]-intercept represents in the equation [tex]\( y = 75x + 50 \)[/tex], where [tex]\( y \)[/tex] is the amount of money in the account and [tex]\( x \)[/tex] is the number of deposits, let's break down the components of the equation.

1. Equation Structure:
- The equation is linear, in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope and [tex]\( b \)[/tex] represents the [tex]\( y \)[/tex]-intercept.

2. Interpreting the Slope and [tex]\( y \)[/tex]-intercept:
- The slope ([tex]\( m \)[/tex]) in this context is 75, which tells us how much money Mario deposits each month.
- The [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is 50, which represents the initial amount of money in the account before any deposits are made.

3. Understanding the Situation:
- At [tex]\( x = 0 \)[/tex] (which means before any deposits), the value of [tex]\( y \)[/tex] would be [tex]\( y = 75 \cdot 0 + 50 \)[/tex]. This simplifies to [tex]\( y = 50 \)[/tex].
- Thus, the [tex]\( y \)[/tex]-intercept shows the starting amount of money in Mario’s account before he begins making monthly deposits.

4. Conclusion:
- Given this, the [tex]\( y \)[/tex]-intercept of 50 means that Mario was given [tex]\( \$50 \)[/tex] for his birthday. This initial amount was placed into his account before he started making regular monthly deposits of [tex]\( \$75 \)[/tex].

Considering these points, the correct interpretation of the [tex]\( y \)[/tex]-intercept is:

B. He was given $50 for his birthday.