Solving modular Diophantine

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This has to do with how cryptography or at least some form of it is implemented.

I'm asked to solve an equation in the form j . k . j= 1 (mod z) z being a "totient" (the result of two primes p and q where z=(p-1)(q-1)) and k a prime number such as z non divisible by k

My prime numbers are p=2, q=3

z= 2

k=5

Back to the equation:j . k . j= 1 (mod z)

I'm told its int the form xy= 1 (mod n)...first I don't see why the original form is like what I'm told, but lets pass on this..(maybe someone can help me understand why later on) According to a rule, we can write 5x+2y =1 in the form of a= c(mod b) so: 5x= 1 (mod 2)

I choose to use mod 5 to isolate x (I'm not sure if I can do that but using the inverse doesn't help)

5(5x)= 5 (mod 2)

x= 5 (mod 2)

x= 5 + 2k

By replacing in 5x+2y =1

5(5 +2k) + 2y =1

25 + 10k +2y =1

2y = -25 -10k +1

2y = -24 -10k

y = -12 -5k

My question...Am I right and if not how to proceed? Thank you in advance for your help.