To solve the equation \( y - \frac{2}{5} = -\frac{1}{3} \) without clearing the fractions first, we'll isolate \( y \) while keeping the fractions in the equation. Here are the steps:
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Start with the original equation: \[ y - \frac{2}{5} = -\frac{1}{3} \]
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To isolate \( y \), we can add \(\frac{2}{5}\) to both sides of the equation: \[ y = -\frac{1}{3} + \frac{2}{5} \]
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To add the fractions \(-\frac{1}{3}\) and \(\frac{2}{5}\), we need a common denominator. The least common multiple (LCM) of \(3\) and \(5\) is \(15\).
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Convert \(-\frac{1}{3}\) to have a denominator of \(15\): \[ -\frac{1}{3} = -\frac{1 \times 5}{3 \times 5} = -\frac{5}{15} \]
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Convert \(\frac{2}{5}\) to have a denominator of \(15\): \[ \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \]
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Now, the equation looks like this: \[ y = -\frac{5}{15} + \frac{6}{15} \]
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Combine the fractions: \[ y = \frac{-5 + 6}{15} = \frac{1}{15} \]
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Thus, the solution to the equation is: \[ y = \frac{1}{15} \]
So the final answer is: \[ y = \frac{1}{15} \]
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