Let's go through the problem step-by-step.
We are given the following information:
1. Nia takes a violin lesson and a dance lesson each week.
2. The dance lesson costs \(\frac{2}{3}\) the cost of the violin lesson.
3. The total cost of both lessons combined is \$75.
We need to find the correct system of equations that can be used to determine \(d\), the cost of the dance lesson in dollars, and \(v\), the cost of the violin lesson in dollars.
Let's start by setting up our variables:
- Let \(v\) be the cost of the violin lesson in dollars.
- Let \(d\) be the cost of the dance lesson in dollars.
From the information given, we can write two key equations:
1. The dance lesson costs \(\frac{2}{3}\) of the cost of the violin lesson. Therefore:
[tex]\[ d = \frac{2}{3} v \][/tex]
2. The total cost of both lessons combined is \$75. Therefore:
[tex]\[ d + v = 75 \][/tex]
Now we have a system of two equations:
[tex]\[ \begin{cases}
d = \frac{2}{3} v \\
d + v = 75
\end{cases} \][/tex]
We need to match this system with one of the given choices.
Choice (A) is:
[tex]\[ \begin{cases}
d = \frac{2}{3} v \\
d + v = 75
\end{cases} \][/tex]
Choice (B) is incorrect because \(d - v = 75\) does not align with the total cost being \$75, and it has an undefined second equation \(u = -\lambda\).
Therefore, the correct choice is:
(A)
[tex]\[ \begin{cases}
d = \frac{2}{3} v \\
d + v = 75
\end{cases} \][/tex]
Thus, the correct system of equations to find \(d\) and \(v\) is:
[tex]\[ \begin{cases}
d = \frac{2}{3} v \\
d + v = 75
\end{cases} \][/tex]
