The best way to solve this problem is by using Simon's Favorite Factoring Trick.
We begin by isolating the coefficient. This gives: \(6mn+2n-m=27\)
Now, factor out \(n\): \(2n(3m+1)-m=27\)
We now need to factor out \(m\), in terms of \((3m+1)\), but, in order to do so, \(m\) must be a multiple of 3.
To achieve this, we multiply the entire equation by 3:\(6n(3m+1)-3m=81\)
Now, we can factor out m as so: \(6n(3m+1) -1(3m+1)=-1+81\)
Simplifying, we get: \((6n-1)(3m+1)=80\)
We now need integer factors, so we try out each of the factor pairs of 80.
The only successful integer solution is \(\color{brown}\boxed{(5, 1)}\)
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