Number theory...

Hi UnVerifiedTaxPayer!

ummm..just a note, but you used different variables in the problem, $a$ and $x$

I'm assuming this is a typo, but if it isn't correct me. Let's just say we are trying to find the value of $2x + 12 \pmod{7}$

Now, since we already have the first equation in handy, we can easily do a simple process.

Let's try to turn the left side of $x \equiv 5 \pmod{7}$ into 2x+12. 

First, let's multiply both sides of the equation by 2. This gets us

$2x \equiv 10 \pmod7$

Since 10 mod 7 is the same as 3 mod 7 (they both leave a remainder of 3), we have the equation $2x \equiv 3 \pmod 7$

Now, let's add 12 to both sides of the equation. We then get $2x+12 \equiv 15 \pmod7$

Since 15 mod 7 is the same as 1 mod 7, we are left with

$2x+12 \equiv 1 \pmod7$

Now, we can check our answer. 

Take 5 as an example. We have that

$5 \equiv 5 \pmod7$

We also have that $2(5) + 12 = 22 \equiv 1 \pmod7$

This also works for 12, 19, and 26, so our answer should be correct. 

So our answer is 1. Again, correct me if I misunderstood the typo in the problem! 

Thanks! :)