To understand what the [tex]\( y \)[/tex]-intercept represents in the equation [tex]\( y = 50x + 75 \)[/tex], we'll break down the equation and examine its components related to Jerry's birthday money and monthly deposits.
The equation is written in the slope-intercept form of a linear equation, which has the general form [tex]\( y = mx + b \)[/tex]. Here, [tex]\( m \)[/tex] represents the slope, and [tex]\( b \)[/tex] represents the [tex]\( y \)[/tex]-intercept.
1. Identify the [tex]\( y \)[/tex]-intercept:
- In the equation [tex]\( y = 50x + 75 \)[/tex], the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is 75. This means that when [tex]\( x = 0 \)[/tex] (which indicates that no deposits have been made yet), the amount of money in the account ([tex]\( y \)[/tex]) is [tex]$75.
2. Interpret the \( y \)-intercept:
- When \( x = 0 \), the equation \( y = 50x + 75 \) simplifies to \( y = 75 \). This value of $[/tex]75 is the initial amount of money in the account before any deposits are made.
- Hence, the [tex]\( y \)[/tex]-intercept of 75 represents the initial money Jerry had in his account before he started making monthly deposits.
3. Analyze the context:
- According to the situation, Jerry was given some birthday money, which he put into the account initially.
- The [tex]\( y \)[/tex]-intercept represents this initial amount, which aligns with the context of being given money as a starting balance.
4. Consider the answer choices:
- A. He puts [tex]$\$[/tex] 75[tex]$ in the account each month. This is incorrect because $[/tex]50x[tex]$ represents the monthly deposits, not $[/tex]75.
- B. He was given [tex]$\$[/tex] 75[tex]$ for his birthday. This is correct. The initial balance of $[/tex]75 represents the money given as birthday money.
- C. He puts [tex]$\$[/tex] 50[tex]$ in the account each month. This is partially correct about the monthly deposits but does not explain the \( y \)-intercept.
- D. He was given $[/tex]\[tex]$ 50$[/tex] for his birthday. This is incorrect as it does not align with the [tex]\( y \)[/tex]-intercept of 75.
Therefore, the correct interpretation of the [tex]\( y \)[/tex]-intercept in this situation is:
B. He was given \$75 for his birthday.
