Can someone tell me if this is okay as a direct proof?

Let x be in Z. Prove that is 7x-8 is even, then x is even.

Proof (direct)

If 7x-8 is even then 7x-8 = 2k.
= 7x = 2k+8
= 7x = 2(k+4), (k+4) is in Z.
= 7x = 2k
= x = 2 (k/7), (k/7) is in Z
= x = 2k, so x is even

I know this can be proven by contrapositive by assuming x is odd first, but I was wondering if this would be how I would do this by direct proof.

Question

Can someone tell me if this is okay as a direct proof?

Let x be in Z.  Prove that is 7x-8 is even, then x is even.

Proof (direct)

If 7x-8 is even then 7x-8 = 2k.
=> 7x = 2k+8
=> 7x = 2(k+4), (k+4) is in Z.
=> 7x = 2k
=> x = 2 (k/7), (k/7) is in Z
=> x = 2k, so x is even

I know this can be proven by contrapositive by assuming x is odd first, but I was wondering if this would be how I would do this by direct proof.

anonymous · Answer

I guess I would have to stop using k the whole way through  . . .

myininaya · Answer

if 7x=2(k+4), then 7x is even 
since 7 is odd then x has to be even