To find the values of \( x \) and \( y \) such that the three fractions \( \frac{6}{14} \), \( \frac{x}{7} \), and \( \frac{24}{y} \) are equivalent, we will find a common value for all fractions.
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Simplify \( \frac{6}{14} \): \[ \frac{6}{14} = \frac{3}{7} \]
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Set up the equations for equivalency: Since \( \frac{6}{14} = \frac{3}{7} \), we can set up the following equation for \( \frac{x}{7} \): \[ \frac{x}{7} = \frac{3}{7} \] From this, we can solve for \( x \): \[ x = 3 \]
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Next, set up the equation for \( \frac{24}{y} \): The second equivalency is: \[ \frac{24}{y} = \frac{3}{7} \] To solve for \( y \), we cross-multiply: \[ 3y = 24 \times 7 \] This simplifies to: \[ 3y = 168 \] Now, divide by 3: \[ y = 56 \]
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Final answers: Thus, the numbers that must replace \( x \) and \( y \) are: \[ x = 3, \quad y = 56 \]
So the final answer is: \[ X = 3, \quad Y = 56 \]