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
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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 \( x \geq 0 \).
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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 \( x.\)
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