To simplify the expression \( \frac{31}{0}x + \frac{1}{6}y + \frac{3}{10}x + \frac{3}{6}y \), we first note that \( \frac{31}{0}x \) is undefined, as division by zero is not permissible in mathematics. This means the expression itself is not valid.
Assuming that you meant to write \( \frac{31}{1}x \) instead of \( \frac{31}{0}x \), we would simplify as follows:
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Combine like terms for \( x \) and \( y \):
- The \( x \) terms are \( 31x \) and \( \frac{3}{10}x \).
- To add these, we need a common denominator:
- The common denominator between 1 (of \( 31x \)) and 10 (of \( \frac{3}{10}x \)) is 10.
- Rewrite \( 31x \) as \( \frac{310}{10}x \).
- Now combine: \[ \frac{310}{10}x + \frac{3}{10}x = \frac{310 + 3}{10}x = \frac{313}{10}x. \]
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Now for the \( y \) terms \( \frac{1}{6}y \) and \( \frac{3}{6}y \):
- Combine these by adding the fractions: \[ \frac{1}{6}y + \frac{3}{6}y = \frac{1 + 3}{6}y = \frac{4}{6}y = \frac{2}{3}y \quad (\text{reduce } \frac{4}{6}). \]
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Now combine the simplified \( x \) and \( y \) terms: \[ \frac{313}{10}x + \frac{2}{3}y. \]
This is the simplified form of the expression (assuming the correction was made). If \( \frac{31}{0}x \) was intended, then the entire expression remains undefined.
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