Direct Proof: Prove that if 5x - 7 is odd, then 9x + 2 is even

[@wiggles7976]

I like the modular arithmetic approach, with everything mod 2. Assume 5x-7=1 => x+1=1 => x=0 => 9x+2 = 0+2 = 0, so 9x+2 is a multiple of 2.

[@_wetmath_]

same

[@lucaterle6537]

How do I know when to use direct proof vs proof by contradiction or proof by cases, etc.? Is there a method for this, or will I just pick it up with experience?

[@EbisaDabalo]

Please did you understand your question bro?

[@devondevon4366]

Since odd times even=even AND an even - odd = odd then for 5x-7 to be odd then 'x' must be even. And if the minuend '5x' is even then the value 5x-7 is odd since and even - odd = odd. Since "9x" is even given that 'x' is even, given that even times odd=even then the value 9x+2 must be even since an even + even=even.

[@thiagovieira9377]

My solution : 5x-7 is odd 9x+2 =5x-7+4x+9 9x+1 =5x-7+4x+8 9x+1=odd+ 4(x+8) 4(x+8)... even number (because is divisible by 2) 9x+1=odd +even 9x+1=2k+1+2m (k and m are no negative integers) 9x+2=2k+2m+2 9x+2=2(k+m+1)=2p (p is a no negative integer) 9x+2 is even

[@AdvayPhadke]

I think I got you all beat: 9(5x-7) = 45x - 63 = 5(9x+2) - 73, is an odd number hence, 5(9x+2) must be an even number, which implies 9x+2 is even.

[@midknightcrisis8612]

Am I missing sth? 5x-7 is odd => 5x is even (bc only an even number minus seven is odd) => x is even (bc only an even number times 5 is even) => 9x is even (same reason) => 9x + 2 is even? Idk there's a way to formalize this in a simple way, but that's basically what things boil down to, right?

[@greenfinmusic5142]

if 5x-7 is odd, then 5x must be even (since if 5x was odd, you'd have odd minus odd, which yields even; therefore contradiction, therefore 5x must be not odd). if 5x is even, that means x must be even (since if x was odd, you'd have odd times odd, which yields odd; therefore contradiction, therefore x must be not odd) if x is even, then 9k must be even, since odd times even equals even. if 9k is even, then 9k+2 must be even, since even + 2 = even. therefore, if 5x-7 is odd, then 9k+2 must be even. QED

[@greenfinmusic5142]

But I don't see how to do this by substituting k's and doing algebra, i.e. Show that if 5x-7=2k+1 (definition of odd) then 9x+2=2k (definition of even) for some integer k. Are those the exact same k's in both instances, meaning we can solve the simultaneous equations for x? (or are they different k's, such as k1 and k2, or k and k'?) if they're the same k's, then subtracting the 2nd equation from the first equation yields -4x-9=1 -4x=10 x=-10/4 Does that tell us anything useful? substituting that x value into the initial statements yields 5(-10/4) - 7 = odd -50/4 - 7 = odd which is a false statement. 9(-10/4) + 2 = even -90/4 + 2 = even which is another false statement. So how does that help us at all? The method of substituting k's appears to deadend in irrelevant failure without correctly answering the question. Where are substitutions of k supposed to lead us, and how?

[@rageprod]

Way cleaner than mine (I have no practice with proofs) 5x-7 = 2k+1, for k in Z 5x = 2k+8 x = (2k+8)/5 y = 9x+2 = 9[(2k+8)/5] + 2 = (18k+82)/5 (which is an integer, because 9x+2 is an integer) 5y = 18k + 82 = 2(9+41) (which is even) 5y is even, but 5 is odd, so y must be even. QED.

[@labiribiri1901]

very hard to follow this proof of yours, I have no practice either. But, I don't understand where you're pulling the extra 8 from, what was the point?

[@rageprod]

@@labiribiri1901 5x-7 = 2k+1 (this is a way of saying 5x-7 is odd, because any odd number can be written as 2k+1 for integer k and every number of the type 2k+1 is odd) then I just add 7 on both sides! 5x-7 +7 = 2k +1 +7 5x = 2k+8 I just want to isolate what x is in terms of an arbitrary integer k so that I can use that information to analyse what 9x+2 (which I called y) looks like. Fortunately the proof came right away.

[@rageprod]

@@labiribiri1901 Just found an easier way: 5x-7 is odd, so 5x must be even, because subtracting an odd number changes parity. 9x+2 can be written as 5x + 4x + 2. 4x is even because it's a multiple of 2, 2 is even, and we've just proved that 5x is even. A sum of even numbers is always even, so 9x+2 must be even, QED.

[@aidanfogleman2060]

Here I was thinking about it graphically, like "both of those functions are neither odd nor even" lmao

[@jennifersilves4195]

Why do you just get to add numbers to one side of an equation? I followed on first watch, then started thinking about it.

[@greense65]

You have to add the 4x + 9 to the 5x - 7 on the right side, because 5x - 7 alone does not equal the 9x + 2 on the left side. Is this the answer to what you are asking?

[@zackariasstrindlund3474]

how is 9x +2 proven to be even? 5x-7 we know is even for a k in (k2+1) , putting that equal to 9x+2 would just show that for a k 9x+2 is odd??

[@wilhufftarkin8543]

Are these still from Spivak's Calculus book?

[@Curby0]

Why didn't my teacher do this lol

[@perfectionist7681]

I also thought the same way you did 😂 when i saw some steps

[@yogesh6308]

Anyone pls suggest a book for "Order relations"" Urgent⚠️

[@affirmajim8510]

5x-7 ≡ 1 mod 2 5x ≡ 8 ≡ 0 mod 2 (5+2•4)x ≡ 9x ≡ 0 mod 2 9x+2 ≡ 2 ≡ 0 mod 2 therefore, (9x+2) is even

[@omnomchomsky]

Proof before looking at video. If 5x - 7 is odd, then 9x + 2 is even. 5x - 7 = 1: (2 * 2 + 1) * x - (2 * 3 + 1) 9x + 2 = 2: (2 * 4 + 1) * x - (2 * 1) In order of 5x - 7 to be odd, x must be even: ( 2 * 2 + 1) * 2(k) - (2 * 3 + 1) 10(k) - (7) = 2(5k) - (2*3) + 1 = 2( 5k - 3) + 1 (2 * 4 + 1) * 2(k) - (2 * 1) 18(k) - 2 = 2(9k -1) 2(5k-3) + 1 is always odd by definition 2k + 1 2(9k -1) is always even by definition 2k Therefor if 5x - 7 is odd, 9x + 2 is even.

[@labiribiri1901]

You lost me at the whole "this is the same as 2k+1". I thought the point of a proof is to make it clear to the audience? Im sorry, but if you can tell me how this isnt a "obviously" moment in your proof im going to assume its not directed to the general audience.

[@veldiax]

They're substituting the value of 5x-7 which is 2k+1

[@rmv2204]

ong

[@hussainfawzer]

♥️

[@FellTheSky]

ok, but earth's still flat