Inverse of a equation with modulo operator

Question

I have this equation:

$y=ax+b \quad \pmod{26}$

where a, b are two parameters. I would like to calculate the inverse of this equation, but I don't know which algebra rules I have to apply. Can you help me?

Answer 1

You can find an inverse equation only if $\gcd(a,26)=1$, then $a$ has an inverse $a^{-1}$ such that$$a\cdot a^{-1}\equiv 1\mod{26}$$therefore$$y\equiv ax+b\mod{26}\iff a^{-1}y\equiv x+a^{-1}b\mod 26\\\iff\\ x\equiv a^{-1}y-a^{-1}b\mod 26$$