How to solve a pair of nonlinear equations using Python?

for numerical solution, you can use fsolve:

http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fsolve.html#scipy.optimize.fsolve

from scipy.optimize import fsolve
import math

def equations(p):
    x, y = p
    return (x+y**2-4, math.exp(x) + x*y - 3)

x, y =  fsolve(equations, (1, 1))

print equations((x, y))