Let's break down the problem step-by-step:
a. We need to write a linear function that represents the remaining money owed [tex]\( L(x) \)[/tex] after [tex]\( x \)[/tex] months. Joey borrows [tex]$2400 from his grandfather and pays back $[/tex]200 monthly.
To create the linear function:
- The initial amount borrowed is [tex]$2400.
- Each month, $[/tex]200 is paid back, which decreases the amount owed.
Thus, the function [tex]\( L(x) \)[/tex] can be written as:
[tex]\[ L(x) = 2400 - 200x \][/tex]
This represents that each month [tex]\( x \)[/tex], the remaining amount decreases by [tex]$200.
b. Now, we need to evaluate \( L(10) \) and interpret its meaning.
Substitute \( x = 10 \) into the linear function:
\[ L(10) = 2400 - 200 \times 10 \]
\[ L(10) = 2400 - 2000 \]
\[ L(10) = 400 \]
This result means that after 10 months, Joey still owes $[/tex]400 to his grandfather.
Now, let's look at the options provided:
a.
[tex]\[
\begin{aligned}
a. & \enspace L(x) = -200x + 2400; \\
b. & \enspace L(10) = 400 \text{, This represents the amount Joey has paid his grandfather after 10 months}.
\end{aligned}
\][/tex]
b.
[tex]\[
\begin{aligned}
a. & \enspace L(x) = 200x + 2400; \\
b. & \enspace L(10) = 4400 \text{. This represents the amount Joey still owes his grandfather after 10 months}.
\end{aligned}
\][/tex]
c.
[tex]\[
\begin{aligned}
a. & \enspace L(x) = -200x + 2400; \\
b. & \enspace L(10) = 400 \text{. This represents the amount Joey still owes his grandfather after 10 months}.
\end{aligned}
\][/tex]
d.
[tex]\[
\begin{aligned}
a. & \enspace L(x) = 200x + 2400; \\
b. & \enspace L(10) = 4400\text{, This represents the amount Joey has paid his grandfather after 10 months}.
\end{aligned}
\][/tex]
From our solutions:
- The correct function is [tex]\( L(x) = -200x + 2400 \)[/tex].
- The correct interpretation of [tex]\( L(10) = 400 \)[/tex] is the amount Joey still owes his grandfather after 10 months.
Therefore, the correct choice is:
c.
[tex]\[
\begin{aligned}
a. & \enspace L(x) = -200x + 2400; \\
b. & \enspace L(10) = 400\text{. This represents the amount Joey still owes his grandfather after 10 months}.
\end{aligned}
\][/tex]
