UpStudy Free Solution:

To determine which of the given Diophantine equations cannot be solved, we need to check whether each equation has integer solutions. This can be done by examining the greatest common divisor (GCD) of the coefficients of \(x\) and \(y\) and checking if it divides the constant term.

1. For the equation \(33x + 14y = 115\):

- The GCD of 33 and 14 is 1.

- Since 1 divides 115, the equation \(33x + 14y = 115\) has integer solutions.

2. For the equation \(14x + 35y = 93\):

- The GCD of 14 and 35 is 7.

- Since 7 does not divide 93, the equation \(14x + 35y = 93\) does not have integer solutions.

Therefore, the equation \(14x + 35y = 93\) cannot be solved.

The correct answer is:

\(14x + 35y = 93\)

Supplemental Knowledge

A Diophantine equation is an equation that requires integer solutions. The general form of a linear Diophantine equation is \(ax + by = c\), where \(a\), \(b\), and \(c\) are integers. A necessary and sufficient condition for the equation to have integer solutions is that the greatest common divisor (gcd) of \(a\) and \(b\) must divide \(c\).

To determine if a given linear Diophantine equation can be solved, we need to check if \(\gcd ( a, b) \) divides \(c\).

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