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# Linear Diophantine Equations

What amounts of money can you make with only $5 and $7 bills? Can you make $23 with those 2 types of bills?

Consider all ordered pairs of integers \((x,y)\) such that \[\frac{5}{x}+\frac{7}{y} = \frac{12}{xy}.\]

The smallest positive integer value of \(x\) in these ordered pairs is \(1,\) since \(x=y=1\) satisfies the equation. What is the second smallest positive integer value of \(x\) in these ordered pairs?

**Hint:** Don't try to list all of the possibilities!

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