2x - 9y + 5z = -1

The smallest coefficient in absolute value is 2

Write each positive integer in terms of its nearest 
smaller multiple of 2

2x - (8+1)y + (4+1)z = -(0+1)

Distribute

2x - 8y - y + 4z + z = -1

Divide every term by 2

x - 4y -  + 2z +  = -

Isolate fractions on the right

x - 4y + 2z =  -  - 

Let that equal to s

 -  -  = s

Multiply through by 2

y - z - 1 = 2s

Solve for y:

        y = 1 + 2s + z

Let z = t

        y = 1 + 2s + t

Substitute in the original:

2x - 9y + 5z = -1

2x - 9(1 + 2s + t) + 5t = -1

2x - 9 - 18s - 9t + 5t = -1

2x - 9 - 18s - 4t = -1

2x = 8 + 18s + 4t

 x = 4 + 9s + 2t

So:

 x = 4 + 9s + 2t
 y = 1 + 2s +  t
 z =           t

Edwin