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.
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.
