To solve the given problem, let's analyze the given equation and find the answers step-by-step.
The equation provided is:
[tex]\[ B = 50 - 0.1n \][/tex]
where [tex]\( B \)[/tex] is the remaining balance in dollars and [tex]\( n \)[/tex] is the number of minutes Marcos has talked.
### Part (a) How much money was on the card when he purchased it?
To determine the initial amount of money on the card when Marcos purchased it, we look at the value of [tex]\( B \)[/tex] when [tex]\( n = 0 \)[/tex] (because no minutes have been talked yet):
[tex]\[ B = 50 - 0.1 \cdot 0 \][/tex]
[tex]\[ B = 50 \][/tex]
So, the initial balance when Marcos purchased the card was \[tex]$50.
This is the y-intercept of the equation because it represents the starting value or the value of \( B \) when \( n = 0 \).
### Part (b) How many minutes will he have talked when he runs out of money?
To find out when Marcos runs out of money, we need to determine when the balance \( B \) becomes 0:
\[ 0 = 50 - 0.1n \]
We solve for \( n \):
\[ 0.1n = 50 \]
\[ n = \frac{50}{0.1} \]
\[ n = 500 \]
Thus, Marcos will have talked for 500 minutes when he runs out of money.
This is the x-intercept of the equation because it represents the point in time (or number of minutes \( n \)) when the remaining balance \( B \) is zero.
### Summary:
a) How much money was on the card when he purchased it?
\[ \$[/tex]50 \]
This is the y-intercept.
b) How many minutes will he have talked when he runs out of money?
[tex]\[ 500 \][/tex] minutes
This is the x-intercept.
