# Verifying my proof: “the equation $2x - 6y = 3$ has no integer solution to $x$ and $y$”

Question:

Prove the equation $$2x - 6y = 3$$ has no integer solution to $$x$$ and $$y$$.

I need to verify my proof I think I did it correctly, but am not fully sure since I don't have solutions in my book. I basically proved by contradiction and assumed there was an integer solution for x or y. I then solved for $$x$$ and $$y$$ in $$2x - 6y = 3$$ getting $$x = 3y + 3/2$$ and $$y = x/3 - 1/2$$ .since both $$x,y$$ are not integers I said it contradicts that $$x$$ or $$y$$ had an integer solution, meaning the original statement was correct. Did I prove this right, or should I redo?

• You're on the right track, but why is $x = 3y + 3/2$ not an integer? Also, your statement that "both $x, y$ are not integers" is not correct: $(3/2, 0)$ is a solution with $y$ integral. – user296602 May 29 '19 at 5:33
• @T.Bongers I think since adding 3/2 will make it a rational number – user675497 May 29 '19 at 5:35
• Some tips for friendly math formatting: math.meta.stackexchange.com/questions/5020/…. – uniquesolution May 29 '19 at 5:37
• ggg.2(x-3y)=3; x-3y integer.Means: Left hand side is even.Hence? – Peter Szilas May 29 '19 at 5:45

Rewind to the point where you say $$x=3y+3/2$$. We rearrange this to $$x-3y=3/2$$, then note that since we have taken $$x$$ and $$y$$ to be integers, $$x-3y$$ is also an integer. But $$3/2$$ is not an integer, a contradiction.

Alternatively, you may write $$2x - 6y = 2(x-3y).$$ Since $$x$$ and $$y$$ are integers, so must be $$x-3y$$. So $$2(x-3y)$$ must be an even integer, clearly being divisible by $$2$$. But $$3$$ is odd.

Your proof, as it stands, is not correct. Some particular issues:

• You get to a point where $$x = 3y + 3/2$$ and $$y = x/3 - 1/2$$ and state that "since both $$x,y$$ are not integers..."; this assumes the conclusion that you're aiming for.

• It's not true that both $$x$$ and $$y$$ are not integers. You could have one of the two be an integer, and the other not be an integer. For example, $$(0, -1/2)$$ and $$(3/2, 0)$$ are both solutions with $$x$$ or $$y$$ integral, but not both.

• Its not convincing that $$3y + 3/2$$ is not an integer (and the point immediately above shows that that claim is not true!).

• Stylistically, your assumptions are not explicitly stated and there is no introduction to the proof. You haven't stated that you're assuming $$(x, y)$$ to be a pair of integers solving a particular equation. Therefore, it's not clear how the contradiction is reached; at a minimum, you need to form the negation of the statement and clearly include assumptions.

So unfortunately, this is not a properly written proof. But it can be fixed without too much work; here's an outline to follow:

1) Introduce the players. Say "We proceed by contradiction. Assume that $$x, y$$ are integers such that $$2x - 6y = 3.$$

2) Isolate one of the variables and get the contradiction. Perhaps "Then $$x = 3y + \frac 3 2$$. Since $$3y$$ is an integer (why?), the sum $$3y + \frac 3 2$$ is not an integer."

3) Tell the reader why this is a problem. "This contradicts the assumption that $$x$$ is an integer."

• yea I know, my actual written proof was much more detailed and better formatted. Im just lazy and quickly wrote a summary haha. But, thanks for pointing out the issues, they are really helpful – user675497 May 29 '19 at 5:43
• @ggg If you're asking a sincere proof-verification question, then it's important to include the actual proof that you want to verify... rather than a "lazy [...] summary." Please don't waste people's time by doing that. – user296602 May 29 '19 at 5:44
• Exactly what made you thinking that this question is not a homework? – peterh - Reinstate Monica Jun 3 '19 at 10:45
• @peterh If you have a problem with my comments on meta, then discuss it there or flag them. Please do not harass me on main with irrelevant nonsense. – user296602 Jun 3 '19 at 10:52
• @T.Bongers I did not harass you, I am just surprised that you seem answering a no-context PSQ. What if the question would be deleted, would it make you happier? – peterh - Reinstate Monica Jun 3 '19 at 10:56