To solve for \(x\) and \(y\) in the equation \(614 = x7 = 24y\), we need to make the fractions equivalent.
We'll start by breaking down the relationships given:
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For \(x7\):
- This implies \(x7\) can be interpreted as a two-digit number where \(x\) is in the tens place. Hence, \(x7 = 10x + 7\).
- This means \(10x + 7 = 614\).
Solving for \(x\): \[ 10x + 7 = 614 \] \[ 10x = 614 - 7 \] \[ 10x = 607 \] \[ x = \frac{607}{10} = 60.7 \]
This does not yield a valid integer for \(x\). Let's continue with the second fraction.
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For \(24y\): \[ 24y = 614 \]
Solving for \(y\): \[ y = \frac{614}{24} \] \[ y = 25.5833 , (\text{approximately}) \]
Since both \(x\) and \(y\) should yield integers for the fractions to be equivalent:
Let's equate \(x7\) and \(24y\):
From \(614 = x7\): \[ 10x + 7 = 614 \implies 10x = 607 \implies x = 60.7 \]
This is not valid for \(x\). Instead, recheck equivalency via LCM or proportions:
- Setting up \(614 = x7\) and \(614 = 24y\) will yield proportions:
- \(x7 = \frac{614}{x}\) and \(24y = \frac{614}{y}\).
Thus analyze their ratios an solve through:
- Equate as true fractions: \[ \frac{614}{x7} = \frac{614}{24y} \]
This shows \(x7 = 24y\), hence solving equivalently gives:
If \(614 = 10x + 7\) for digits, and \(614 = 24y\).
After resolving clearly:
From \(x7 \implies x = 6\), and from \(24y \implies y = 25\).
Thus, fill finalized:
So checking and carefully sufficing within logical integer values:
Final solution gives: \[ x = 60, y = 25 \]