To find the number of beads on Audrina's bracelet at the specified minutes, we can use the equation provided:
\[ y = 48 - 4x \]
Let's calculate the number of beads for each given minute:
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For \( x = 2 \): \[ y = 48 - 4(2) = 48 - 8 = 40 \] So, for 2 minutes, Audrina has 40 beads.
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For \( x = 4 \): \[ y = 48 - 4(4) = 48 - 16 = 32 \] So, for 4 minutes, Audrina has 32 beads.
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For \( x = 8 \): \[ y = 48 - 4(8) = 48 - 32 = 16 \] So, for 8 minutes, Audrina has 16 beads.
Now we can complete the table with the number of beads on Audrina's bracelet:
| Minutes (x) | Beads (y) | |-------------|-----------| | 2 | 40 | | 4 | 32 | | 8 | 16 |
Next, to determine if there is a time when Greg and Audrina's bracelets have the same number of beads, we can set up the equations for both bracelets:
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Greg's number of beads after \( x \) minutes is: \[ g = 24 + 2x \]
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Audrina's number of beads after \( x \) minutes is: \[ a = 48 - 4x \]
We want to find \( x \) such that: \[ 24 + 2x = 48 - 4x \]
Now we can solve this equation:
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Combine like terms: \[ 2x + 4x = 48 - 24 \] \[ 6x = 24 \]
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Divide by 6: \[ x = 4 \]
At \( x = 4 \) minutes, both Greg and Audrina will have the same number of beads.
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For Greg at 4 minutes: \[ g = 24 + 2(4) = 24 + 8 = 32 \]
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For Audrina at 4 minutes: \[ a = 48 - 4(4) = 48 - 16 = 32 \]
Conclusion: Yes, there is a time at which both bracelets will have the same number of beads—this occurs at 4 minutes, with both having 32 beads.
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