To find the missing fractions in the equation \(\frac{x}{5} - \frac{1}{2} = \frac{3}{y}\), we can start by isolating \(\frac{x}{5}\).
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Add \(\frac{1}{2}\) to both sides of the equation: \[ \frac{x}{5} = \frac{3}{y} + \frac{1}{2} \]
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To combine the fractions on the right side, we need a common denominator. The least common multiple of \(y\) and \(2\) is \(2y\).
Rewrite \(\frac{3}{y}\) with the common denominator: \[ \frac{3}{y} = \frac{3 \cdot 2}{y \cdot 2} = \frac{6}{2y} \]
Now rewrite \(\frac{1}{2}\): \[ \frac{1}{2} = \frac{1 \cdot y}{2 \cdot y} = \frac{y}{2y} \]
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Now we can combine the fractions: \[ \frac{x}{5} = \frac{6}{2y} + \frac{y}{2y} = \frac{6 + y}{2y} \]
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Now let's clear the fraction by multiplying both sides by \(5 \cdot 2y\): \[ 2yx = 5(6 + y) \]
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Now, expand the right side: \[ 2yx = 30 + 5y \]
At this point, we have a relationship between \(x\) and \(y\). To find specific values, let's assign a value to \(y\) and solve for \(x\).
Let’s use \(y = 6\) as an example: \[ 2x(6) = 30 + 5(6) \] \[ 12x = 30 + 30 \] \[ 12x = 60 \] \[ x = \frac{60}{12} = 5 \]
Now substituting these values back into the equation: \[ \frac{5}{5} - \frac{1}{2} = \frac{3}{6} \] This simplifies to: \[ 1 - \frac{1}{2} = \frac{1}{2} \] This is true.
Thus, the missing fractions are: \[ x = 5 \quad \text{and} \quad y = 6. \]
So the filled fractions are: \[ \frac{5}{5} - \frac{1}{2} = \frac{3}{6}. \]
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