elementary number theory - Euclid Algorithm to Find Muliplicative Inverse

Here I am trying to find the multiplicative inverse of 19 respect to 29.

$$19x \equiv 1 \pmod{29}$$

What I tried

$$\begin{align*} 29 &= 1(19) + 10\\\ 19 &= 1(10) + 9\\\ 10 &= 1(9) + 1. \end{align*}$$

From backtracking, I came up with the

$$\begin{align*} 1 &= 2(29) - 3(19)\\\ \end{align*}$$

However, 3 is not a multiplicative inverse of the 29. Where am I making a mistake?

I looked many answers including this answer; however, couldn't figure out my mistake.

Answer

What you have found indeed is that $-3\equiv 26$ is the multiplicative inverse of $19$ $\mod 29$.