Emma starts a piano class that costs $60 per week. Six weeks later, she starts a dance class that costs $25 per week. Part A: Enter an equation that represents the total cost in d... Emma starts a piano class that costs $60 per week. Six weeks later, she starts a dance class that costs $25 per week. Part A: Enter an equation that represents the total cost in dollars y for both classes x weeks after starting the piano class. Part B: If Emma takes piano classes for 10 weeks, what is the total cost for both classes? Read more

Steps to Solve

1. Define the cost of the piano class

The cost of the piano class is $60 per week. So after $x$ weeks, the cost will be $60x$.

2. Define the cost of the dance class

The dance class starts 6 weeks later and costs $25 per week. Therefore, the number of weeks Emma attends dance class is $x - 6$. The cost of the dance class is $25(x-6)$. Note that this is only true when $x > 6$.

3. Define the equation for the total cost

The total cost, $y$, is the sum of the piano and dance class costs. Considering that the dance class only starts after 6 weeks we split this into two cases:

If $x \le 6$, $y = 60x$

If $x > 6$, $y = 60x + 25(x-6)$

4. Simplify the equation for $x > 6$

Simplify the equation $y = 60x + 25(x-6)$

$y = 60x + 25x - 150$ $y = 85x - 150$

5. Write the complete equation for y

$y = \begin{cases} 60x, & x \le 6 \ 85x - 150, & x > 6 \end{cases}$

6. Answer part A

The equation that represents the total cost in dollars $y$ for both classes $x$ weeks after starting the piano class is:

$y = \begin{cases} 60x, & x \le 6 \ 85x - 150, & x > 6 \end{cases}$

7. Calculate the total cost for 10 weeks of piano classes

Since 10 > 6, substitute $x = 10$ into the equation $y = 85x - 150$.

$y = 85(10) - 150$ $y = 850 - 150$ $y = 700$