Let us first look at the conventional approach.
7x - 5y = 3
$\Rightarrow$ 7x = 5y+3 ----------------(1)
and
-7x + 5y = 3
$\Rightarrow$ 7x+3 = 5y ----------------(2)
Solving equation (1), we get the first integral value for y, at which x is an integer at y = 5. Values of y will increase in steps of 7 (the coefficient of x). The next few values of "y" satisfying the equation will be 5,12,19.......96.
Number of terms = 13 (considering 2 digit numbers)
Solving equation (2), we get the first integral value for y at y = 2. Values of y will increase in steps of 7. Hence the second AP will be 2, 9, 16, 23...... 93.
Number of terms = 12 (considering 2 digit numbers)
Number of values of y which satisfy this equation = 25
Therefore, number of values which do not satisfy this equation = 90 - 25 = 65. The answer is option (b).

Shortcut:
We know that there are 90 2-digit numbers.
The values which satisfy for "y" form an AP with a common difference = 7 (the coefficient of x).
Hence, the number of terms in that AP $=\frac{90}{7}\times 2=24or25$; since there are two APs.
The answer has to be either 90-24 = 66 or 90-25 = 65. Answer is option (b).