Each week, Nia takes a violin lesson and a dance lesson. The dance lesson costs \(\frac{2}{3}\) as much as the violin lesson, and the cost of both lessons combined is \$75. Which of the following systems of equations could be used to find \(d\), the cost of the dance lesson in dollars, and \(v\), the cost of the violin lesson in dollars?

A.
[tex]\[ \begin{cases}
d = \frac{2}{3}v \\
d + v = 75
\end{cases} \][/tex]

B.
[tex]\[ \begin{cases}
d = \frac{3}{2}v \\
d + v = 75
\end{cases} \][/tex]

C.
[tex]\[ \begin{cases}
v = \frac{2}{3}d \\
d + v = 75
\end{cases} \][/tex]

D.
[tex]\[ \begin{cases}
v = \frac{3}{2}d \\
d + v = 75
\end{cases} \][/tex]



Answer :

To solve for \( d \) (the cost of the dance lesson) and \( v \) (the cost of the violin lesson) given the conditions, we will set up a system of equations based on the problem statement.

1. Define the relationship between the costs of the dance and violin lessons:
- We are told that the dance lesson costs \(\frac{2}{3}\) as much as the violin lesson. This can be written as:
[tex]\[ d = \frac{2}{3} v \][/tex]

2. Sum of the costs:
- We are also told the total cost of both lessons combined is $75. This gives us:
[tex]\[ d + v = 75 \][/tex]

Now we have the two equations:
[tex]\[ \begin{cases} d = \frac{2}{3} v & \text{(1)} \\ d + v = 75 & \text{(2)} \end{cases} \][/tex]

This system of equations can be used to find \( d \) and \( v \).

Solving the system of equations:

First, substitute equation (1) into equation (2):

[tex]\[ \left( \frac{2}{3} v \right) + v = 75 \][/tex]

Combine the terms involving \( v \):

[tex]\[ \frac{2}{3} v + v = 75 \][/tex]

Convert \( v \) to a common fraction:

[tex]\[ \frac{2}{3} v + \frac{3}{3} v = 75 \][/tex]

Simplify:

[tex]\[ \frac{5}{3} v = 75 \][/tex]

To isolate \( v \), multiply both sides of the equation by \( \frac{3}{5} \):

[tex]\[ v = 75 \times \frac{3}{5} \][/tex]

So, we get:

[tex]\[ v = 45 \][/tex]

Next, use this value to find \( d \). From equation (1):

[tex]\[ d = \frac{2}{3} \times 45 \][/tex]

Thus,

[tex]\[ d = 30 \][/tex]

Conclusion:

The system of equations to determine the costs of the dance lesson (\( d \)) and the violin lesson (\( v \)) is:

[tex]\[ \begin{cases} d = \frac{2}{3} v \\ d + v = 75 \end{cases} \][/tex]

And the solution to this system is:
[tex]\[ v = 45, \quad d = 30 \][/tex]