Question

Taylor is comparing the success of these two songs. She wants to write an equation that would show how many days it would take for the two songs to have the exact same number of streams. She used x for the days that have passed. Which equation is correct?
Using your answer for Question 3, indicate how many solutions the equation in Question 3 would have. Then indicate how many days would have to pass for each song to have the same exact number of streams. In other words, solve the equation in Question 3.

Answer 1

To help Taylor compare the success of the two songs, we need some specifics about the number of streams each song starts with and the rate at which each song gains streams over time.

Let's say:

- Song A starts with \( S_A \) streams and gains \( R_A \) streams per day.
- Song B starts with \( S_B \) streams and gains \( R_B \) streams per day.

The equation representing the number of streams after \( x \) days would be:

For Song A:
\[ S_A + R_A \cdot x \]

For Song B:
\[ S_B + R_B \cdot x \]

To find out when both songs will have the same number of streams, Taylor can set the two expressions equal to each other:

\[ S_A + R_A \cdot x = S_B + R_B \cdot x \]

Rearranging the equation gives:

\[ R_A \cdot x - R_B \cdot x = S_B - S_A \]

Factoring out \( x \):

\[ (R_A - R_B) \cdot x = S_B - S_A \]

To solve for \( x \):
\[ x = \frac{S_B - S_A}{R_A - R_B} \]

### Determining the Solutions

1. **Number of Solutions**: The equation will have one solution if \( R_A \neq R_B \) (i.e., if the growth rates are different). If \( R_A = R_B \), then the two songs will never equal each other unless they started with the same number of streams (\( S_A = S_B \)), in which case there are infinitely many solutions for any \( x \geq 0 \).

2. **Number of Days**: To find out how many days must pass for the two songs to have the same number of streams, you would calculate \( x \) using the equation derived earlier. If the denominator \( (R_A - R_B) \) is non-zero, you'll have a specific solution for \( x \). If it is negative or zero, then you interpret it based on the context.

So, to summarize, please provide the specific values for \( S_A, R_A, S_B, \) and \( R_B \) if you want a numerical solution for \( x.\)