To complete the table for the number of beads on Ezra's bracelet using the equation \( y = 48 - 4x \), we will substitute the values of \( x \) (representing minutes) into the equation.
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For \( x = 2 \): \[ y = 48 - 4(2) = 48 - 8 = 40 \]
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For \( x = 4 \): \[ y = 48 - 4(4) = 48 - 16 = 32 \]
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For \( x = 8 \): \[ y = 48 - 4(8) = 48 - 32 = 16 \]
Now, we can fill in the table:
| Minutes (x) | Beads (y) | |--------------|-----------| | 2 | 40 | | 4 | 32 | | 8 | 16 |
Now, to determine if there is a time when Nancy's and Ezra's bracelets will have the same number of beads, we can set up the equations for both:
For Nancy: \[ N(x) = 24 + 2x \]
For Ezra: \[ E(x) = 48 - 4x \]
We want to find \( x \) when \( N(x) = E(x) \): \[ 24 + 2x = 48 - 4x \]
Now, we will solve for \( x \):
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Add \( 4x \) to both sides: \[ 24 + 2x + 4x = 48 \] \[ 24 + 6x = 48 \]
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Subtract \( 24 \) from both sides: \[ 6x = 48 - 24 \] \[ 6x = 24 \]
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Divide by \( 6 \): \[ x = \frac{24}{6} = 4 \]
Now, let's check if they have the same number of beads at \( x = 4 \):
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Nancy's beads: \[ N(4) = 24 + 2(4) = 24 + 8 = 32 \]
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Ezra's beads: \[ E(4) = 48 - 4(4) = 48 - 16 = 32 \]
Yes, at \( x = 4 \) minutes, both Nancy and Ezra have 32 beads. Therefore, there is a time when their bracelets will have the same number of beads, and that time is 4 minutes.