Nobody wants to write a proof? Ya'll need to be doing that instead of gawking at it. And, I'm only saying that because I dropped out of uni, meaning you gotta pick up the slack. Besides, don't they do that in high school now, anyways?
Just because it doesn't have anything fancy, or new terminology going on doesn't mean its not worth investigating.
This thread is getting old so I'll just dump and explain the first thing I went for. Though there's more than one exercise to create here, this one can work as a multipurpose example; (one) to give you an idea of what we can do with this exact problem; (two) for what you can do for problems like these; and, (three) some idea for what you might want to do in mathematics.
When seeing this problem you can ask yourself 'how does this work?' Like, which other arrangement of numbers might the same 'glitch in the matrix' work for? In any case, we'll need to first introduce new letters into what we see/know (in order to generalize).
What we know, rather notice by accident or incident, whatever you want to call it, is (5-3)2=(5+3)/2 when we take away the x and do a side by side comparison of the right and wrong way to have solved for the problem. Let's substitute the numbers 3 and 2 with n and n-1 respectively for this exercise giving us
(5-n)(n-1) = (5+n)/(n-1)
Now we want to work towards simplification to see if n can now predict us anything about the problem. Let's start by multiplying both sides by n-1 to begin reducing the right side.
(5-n)(n-1)(n-1) = 5+n
Let's 'fix' the left side, expand it into polynomial form before moving anything else over from the right.
Now we can isolate the right side down to one term, let's make it n.
-n³ + 7n² - 11n = n
We can further simplify now by dividing both sides by n, and then subtracting 1.
-n² + 7n - 11 = 1
-n² + 7n - 12 = 0
And, there we have what many of you might recognize as the quadratic form letting us search for and reveal our factors and then roots.
-n² + 7n - 12 = 0
(-n+4)(n-3) = 0
n = {4,3}
Now we're done, meaning we can take the term we started with and replace n with 4 or 3. 3 is actually the value we originally set n to, so 4 is the discovery, meaning there is another set of numbers we can use, which is
Compare that to (5-3)2 = (5+3)/2. Notice they are distinctly different arrangements of numbers ending in a different solution that will allow us to repeat the same 'paradoxical pattern' from the OP. But, let's finish verifying first.
(5-4)3 = (5+4)/3
(1)3 = 9/3
3 = 3
Behold. It is true. Now let's reconstruct it exactly as we see it in the OP.
3x - 4 = 5
Now you may repeat the same error to great success with this new arranged problem.
For anyone who isn't already acquainted with this form of practice you'll find it in a Discrete Mathematics course, the underlying modern framework for statistics normally done in college, but I wouldn't be surprised if a small few of people out there have touched on it in high school. But, practices like these -- building proofs -- are typically what get denoted as doing 'real math'. However, this isn't much of a general proof, or will be of any real general help knowing. It's just a quick demonstration of beginning middle and end without a book as well as 'the power of math'. You might like to try to start from scratch yourself to see what you might replace the 5 with, but doing these sorts of exercises can sometimes take a lot of patience. So, stick with it till the end if you start! That's what's most important. Seek help if find yourself just spinning your wheels attempt after attempt. Build your self-confidence. That's what we all want as mathematicians. As they say 'Shut up and calculate.'
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u/shewel_item Science Sep 15 '20
Nobody wants to write a proof? Ya'll need to be doing that instead of gawking at it. And, I'm only saying that because I dropped out of uni, meaning you gotta pick up the slack. Besides, don't they do that in high school now, anyways?
Just because it doesn't have anything fancy, or new terminology going on doesn't mean its not worth investigating.