To solve this problem, we need to establish a system of equations that correctly represents the given conditions:
1. The total number of DVDs is 15.
2. The total cost of the DVDs is \[tex]$156.
3. There are two types of DVDs: one type costs \$[/tex]11 each, and the other costs \[tex]$9 each.
Let's denote:
- \( x \) as the number of \$[/tex]11 DVDs,
- [tex]\( y \)[/tex] as the number of \[tex]$9 DVDs.
From the information provided, we can establish the following equations:
1. The total number of DVDs is 15:
\[
x + y = 15
\]
2. The total cost of the DVDs is \$[/tex]156. Since each \[tex]$11 DVD contributes \$[/tex]11 to the total cost, and each \[tex]$9 DVD contributes \$[/tex]9 to the total cost, we can write:
[tex]\[
11x + 9y = 156
\][/tex]
Thus, the correct system of equations to determine [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
\begin{cases}
x + y = 15 \\
11x + 9y = 156
\end{cases}
\][/tex]
The answer corresponds to option D:
[tex]\[
x + y = 15 \\
11x + 9 y=156
\][/tex]
