To determine the meaning of the [tex]\( y \)[/tex]-intercept in the equation [tex]\( y = 75x + 60 \)[/tex], let's break down the components of the equation:
1. [tex]\( y \)[/tex] represents the total amount of money in the account after [tex]\( x \)[/tex] months.
2. [tex]\( x \)[/tex] is the number of months (or number of deposits made).
3. The coefficient [tex]\( 75 \)[/tex] indicates how much money Karma deposits each month.
4. The [tex]\( y \)[/tex]-intercept is the constant term [tex]\( 60 \)[/tex] in the equation.
The [tex]\( y \)[/tex]-intercept of an equation is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. This represents the initial amount of money in the account before any deposits are made.
Therefore, when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 75(0) + 60 = 60. \][/tex]
So, when no deposits have been made yet (i.e., [tex]\( x = 0 \)[/tex]), the amount of money in the account is [tex]$60. This $[/tex]60 is the initial amount she had in the account, which is the birthday money she was given.
Thus, the [tex]\( y \)[/tex]-intercept corresponds to the initial amount of money she had in the account before making any deposits.
The correct interpretation of the [tex]\( y \)[/tex]-intercept in this context is:
D. She was given [tex]$\$[/tex] 60$ for her birthday.
