Al+Fe2O4->Fe+Al2O3

2Al+Fe2O3->2Fe+Al2O3

40Al+20Fe2O3->40Fe+20Al2O3

Nope!

Pb->Au

C7H16+O2->CO2+H2O
C7H16+11O2->7CO2+8H2O

Al+Fe2O3->Fe+Al2O3
2Al+Fe2O3->2Fe+Al2O3

Pb->Au
Nope!

// element use table, then once parsed reused as molecule weights
u,t[99];

// molecules
char*s,*m[99]; // name and following separator
c,v[99][99]; // count-1, element vector

i,j,n;

// brute force solver, n==0 upon solution - assume at most 30 of each molecule
b(k){
    if(k<0)for(n=j=0;!n&&j<u;j++)for(i=0;i<=c;i++)n+=t[i]*v[i][j]; // check if sums to zero
    else for(t[k]=0;n&&t[k]++<30;)b(k-1); // loop through all combos of weights
}

main(int r,char**a){
    // parse
    for(s=m[0]=a[1];*s;){
        // parse separator, advance next molecule
        if(*s==45)r=0,s++;
        if(*s<65)m[++c]=++s;
        // parse element
        j=*s++;
        if(*s>96)j=*s+++j<<8;
        // lookup element index
        for(i=0,t[u]=j;t[i]-j;i++);
        u+=i==u;
        // parse amount
        for(n=0;*s>>4==3;)n=n*10+*s++-48;
        n+=!n;
        // store element count in molecule vector, flip sign for other side of '->'
        v[c][i]=r?n:-n;
    }
    // solve
    b(c);
    // output
    for(i=0,s=n?"Nope!":a[1];*s;putchar(*s++))s==m[i]&&t[i++]>1?printf("%d",t[i-1]):0;
    putchar(10);
}

./a.out "C7H16+O2->CO2+H2O"
./a.out "Al+Fe2O4->Fe+Al2O3"
./a.out "Pb->Au"

C7H16+11O2->7CO2+8H2O
8Al+3Fe2O4->6Fe+4Al2O3
Nope!

b@t_ :=Quiet@Check[Module[{s = StringSplit[t, "+" | "->"], g = StringCases, k = Length,
  e, v, f, z, r},
e = Union@Flatten[g[#, _?UpperCaseQ ~~ ___?LowerCaseQ] & /@ s];v = k@e;
s_~f~e_ := If[g[s, e] == {}, 0, If[(r = g[s, e ~~ p__?DigitQ :> p]) == {}, 1,
   r /. {{x_} :> ToExpression@x}]];z = k@s - v;
r = #/(GCD @@ #) &[Inverse[Join[SparseArray[{{i_, j_} :> f[s[[j]], e[[i]]]}, k /@ {e, s}],
Table[Join[ConstantArray[0, {z, v}][[i]], #[[i]]], {i, k[#]}]]][[All, -1]] &
   [IdentityMatrix@z]];
Row@Flatten[ReplacePart[Riffle[Partition[Riffle[Abs@r, s], 2], " + "],
   2 Count[r, _?Negative] -> " -> "]]], "Nope!"]

b["C7H16+O2->CO2+H2O"]
b["Al+Fe2O3->Fe+Al2O3"]
b["Pb->Au"]

import sys,re
from sympy.solvers import solve
from sympy import Symbol
from fractions import gcd
from collections import defaultdict

Ls=list('abcdefghijklmnopqrstuvwxyz')
eq=sys.argv[1]
Ss,Os,Es,a,i=defaultdict(list),Ls[:],[],1,1
for p in eq.split('->'):
 for k in p.split('+'):
  c = [Ls.pop(0), 1]
  for e,m in re.findall('([A-Z][a-z]?)([0-9]*)',k):
   m=1 if m=='' else int(m)
   a*=m
   d=[c[0],c[1]*m*i]
   Ss[e][:0],Es[:0]=[d],[[e,d]]
 i=-1
Ys=dict((s,eval('Symbol("'+s+'")')) for s in Os if s not in Ls)
Qs=[eval('+'.join('%d*%s'%(c[1],c[0]) for c in Ss[s]),{},Ys) for s in Ss]+[Ys['a']-a]
k=solve(Qs,*Ys)
if k:
 N=[k[Ys[s]] for s in sorted(Ys)]
 g=N[0]
 for a1, a2 in zip(N[0::2],N[1::2]):g=gcd(g,a2)
 N=[i/g for i in N]
 pM=lambda c: str(c) if c!=1 else ''
 print '->'.join('+'.join(pM(N.pop(0))+str(t) for t in p.split('+')) for p in eq.split('->'))
else:print 'Nope!'

python chem-min.py "C7H16+O2->CO2+H2O"
python chem-min.py "Al+Fe2O4->Fe+Al2O3"
python chem-min.py "Pb->Au"

C7H16+11O2->7CO2+8H2O
8Al+3Fe2O4->6Fe+4Al2O3
Nope!

from sympy import*
import sys,re
from sympy.solvers import*
from collections import*
P=str.split
L=map(chr,range(97,123))
q=sys.argv[1]
S,O,a,i,u,v=defaultdict(list),L[:],1,1,'+','->'
w=u.join
for p in P(q,v):
 for k in P(p,u):
     c=L.pop(0)
     for e,m in re.findall('([A-Z][a-z]*)(d*)',k):
      m=int(m or 1)
      a*=m
      S[e][:0]=[c,m*i],
 i=-1
Y=dict((s,Symbol(s))for s in set(O)-set(L))
Q=[eval(w('%d*%s'%(c[1],c[0])for c in S[s]),{},Y)for s in S]+[Y['a']-a]
k=solve(Q,*Y)
if k:
 N=[k[Y[s]]for s in sorted(Y)]
 g=gcd(N[:1]+N[1::2])
 print v.join(w((lambda c:str(c)*(c!=1))(N.pop(0)/g)+str(t)for t in P(p,u))for p in P(q,v))
else:print'Nope!'