To solve the equation \( 32x - 12y = 9 \) in base 10, we can express one variable in terms of the other. Let’s solve for \( y \) in terms of \( x \).
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Start with the original equation:
\[ 32x - 12y = 9 \]
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Rearranging gives us:
\[ 12y = 32x - 9 \]
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Now divide by 12 to solve for \( y \):
\[ y = \frac{32x - 9}{12} \]
This gives you the relationship between \( x \) and \( y \).
If you'd like to find specific integer solutions, you can substitute various integer values for \( x \) and calculate the corresponding \( y \) values.
For example:
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Let \( x = 0 \):
\[ y = \frac{32(0) - 9}{12} = -\frac{9}{12} = -\frac{3}{4} \quad \text{(not an integer)} \]
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Let \( x = 1 \):
\[ y = \frac{32(1) - 9}{12} = \frac{32 - 9}{12} = \frac{23}{12} \quad \text{(not an integer)} \]
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Let \( x = 2 \):
\[ y = \frac{32(2) - 9}{12} = \frac{64 - 9}{12} = \frac{55}{12} \quad \text{(not an integer)} \]
Continuing this way will allow you to find integer solutions if they're available or you can specify further conditions to limit \( x \) or \( y \).
If you want to find integer solutions specifically or need additional assistance, please let me know!
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