Thanks for the puzzle.
As per parity, in an equation system in the format of x + y = z, there has to be two odd numbers or no odd numbers at all i.e.; either two on the left hand side or one on the left hand side and one on the solution side. But there can never be odd number of odd numbers in an equation like this since odd + even = odd and odd + odd = even. Hence, odd numbers should appear in pairs or not at all.
Since, the total number of odd numbers in 1 - 18 are 9 (which itself is an odd number), there will be always one odd number extra which we cannot put in place.
Hence, the solution to the system is not possible.