We must bring into our attention the chemical formula of hydrazine, N2H4. The products that come out from decomposition reaction of hydrazine are nitrogen,hydrogen and ammonia.
The decomposition reaction of hydrazine that we must balance is:
aN2H4 -> bNH3 + cN2
We must find the coefficients a,b,c that balance the equation, meaning that the number of atoms that enter in your equation must remain unchanged after the reaction of decomposition is over.
We notice that we enter with 4 hydrogen atoms but we come out with only 3, so we calculate the greatest common factor and we get 12. Put a = 3 and b = 4 and then watch what happens:
3N2H4->4NH3 + cN2
Calculate how many atoms of nitrogen come in and come out in equation. 3*2 = 4+2 = 6. It is correct, so let c = 1.
The balanced reaction of decomposition:
3N2H4->4NH3 + N2