Let $$$x$$$ — value, $$$M$$$ — mod, $$$y$$$ — $$$x^{-1}\,(mod\,M)$$$.

where $$$k$$$ is integer

You can see this is just Diophantine equation with coefficients $$$a=x$$$, $$$b=M$$$, $$$c=1$$$.

So we can use extended euclidean algorithm.

def gcd_ext(a, b):
    if b == 0:
        return a, 1, 0
    d, x, y = gcd_ext(b, a % b)
    x, y = y, x - (a // b) * y
    return d, x, y


x = 7
mod = int(1e9)
d, inv, k = gcd_ext(x, mod)
#   d is gcd(x, mod), if d isn't 1, then inverse is non exist
print(inv % mod, k)