Solving linear equivalence mod $26$

Question

I am looking to solve for $x$ in this modular arithmetic problem. We haven't learned anything remotely as complex as this in my class so not exactly sure where to start.

$$y \equiv 5x + 25 \pmod{26}$$

Here's what I know so far.

The additive inverse of 25 in mod 26 is 1 since: $1 + 25 \equiv 0 \pmod{26}$. So, I added 1 to both sides of the congruence.

The multiplicative inverse of 5 in mod 26 is −5 since: $5(−5) \equiv −25 \equiv 1 \pmod{26}$. So, I multiplied by -5 on both sides of the congruence.

After doing those steps this is what I have come up with

$$-5 - 5y \equiv ( -25x - 130) \pmod{26} \tag 1$$

Then I add the 5 over to the other side of the equation to get

$$-5y \equiv -25x - 125 \pmod{26} \tag 2$$

Then I divided by a factor of 5 to each of the numbers

$$-y \equiv -5x - 25 \pmod{26} \tag 3$$

Now I feel stuck and like I'm just back at the start... hmm is this even correct? Let me know where I went wrong or what I need to continue doing! Thanks in advance

Answer 1

$-5 \cdot 5 = -25 \equiv 1 \pmod{26}$ so that the inverse of $5$ is $-5$ modulo $26$.

So if $y\equiv 5x+25 \pmod{26}$ and I want to express $x$ in terms of $y$:

$$y+1 \equiv 5x+26 \pmod{26} \equiv 5x \pmod{26}$$ so

$$-5(y+1) = -5 \cdot 5 x \pmod{26} \equiv x \pmod{26}$$ or rewritten:

$$x \equiv -5y -5 \pmod{26} \equiv 21y + 21 \pmod{26}$$