Why Adding Two Equations To Each Other Works In Solving a System of Linear Equations

 A long time ago, when I was taking algebra in high school, I wondered why you can add two equations to each other to assist you in solving a system of linear equations, but I never really looked into it.

But the other day I thought about it some and I realized why it works.

Say you have two equations:  3x+7y = 4 and 5x-2y = 6.

Now you can add something to both sides of an equation without changing it.  Pretty much whatever you want.  So let's add (5x-2y) to both sides of 3x+7y = 4.

We get 3x+7y + (5x-2y) = 4 + (5x-2y)

BUT the key is, we also KNOW that 5x-2y = 6, so we can just replace (5x-2y) on one side of the equation with 6.

And we get this: 3x+7y + (5x-2y) = 4 + (6)

Which is just adding the two equations together.  (3x+7y = 4) + (5x-2y = 6).

So really when you add two equations together, you're adding the same thing to both sides and then doing a substitution.  And adding the equations directly is just a shortcut.  But a lot of times it's hard to see how the shortcuts work when you don't do them the long way.

And of course it's probably more common to add a multiple of one equation to another rather than to add them together directly, but the idea is still the same.

What is the answer in this example?  I have no clue.  But as Tom Lehrer once said, the important thing is to know what you're doing rather than to get the right answer.  That's probably why a lot of grad school textbooks don't give you the answers to most of the worked problems.

Stumping AI

The best way to stump AI is to ask it really obscure questions.  One time I asked it about a WWF house show that took place at the Summit in Houston in 1997 shortly after Summerslam that year.  I asked it to guess not what the main event of that show actually WAS, but what the main event of the show was initially ADVERTISED to be.  It said Shawn Michaels vs. Undertaker.  Not too bad of a guess because the first time those guys ever faced each other was very close to being around that time.

But the actual answer to that question was Stone Cold vs. Undertaker vs. Bret Hart in a triple threat match.  That was the advertised main event that I saw on a flyer in a store, and I remember being super excited to see it.  But unfortunately, Stone Cold was severely injured at Summerslam and could not perform at the following house show in Houston at all.  So, the main event ended up being Bret Hart vs the Undertaker, which I still enjoyed since I didn't see too many events live.  

What's sad is not only did that match not happen at that event, but it never ended up happening period.  By the time Stone Cold recovered enough from his injury to perform again, Bret Hart left the company and then retired shortly after.

If I asked AI that question again, maybe it learned the answer from me and would get the question right next time.  Or perhaps it would pull up and reference the text I'm writing now and try to use that as an answer.  Who knows.  I imagine it would have to trust my credibility.  But I remember that flyer for that triple threat match plain as day.  Even if everyone forgot it and there's no copies left, that flyer advertising Stone Cold vs. Undertaker vs. Bret Hart in Houston in 1997 really existed.  As Kurt Angle would say, it's true.  It's true.

Easy Proof That the Square Root of a Prime is Irrational

Fun fact I discovered the other day.  I'm surprised I wasn't aware of it before.

Take any whole number.  Divide it by another whole number satisfying the condition that you have a fraction that may be improper but otherwise cannot be reduced.  (The two numbers in the fraction are relatively prime.)  

Multiply this fraction by itself.  If the bottom number of the fraction is larger, squaring the fraction will make the bottom number grow more than the top number, so you can't get a whole number by squaring this fraction.  A similar argument can be made if the top number is larger.  There's no way to get a whole number squaring the fraction in that case either.  

Since you cannot get a whole number by squaring a fraction, you can't have a fraction of this kind by taking the square root of a whole number.  So the only kind of number a square root of a whole number can be is either another whole number, or an irrational.  The square root of a prime cannot be a whole number since it has no whole number factors, so the square root of a prime is irrational.

Enough

 I don't necessarily want equal.  I just want enough.

Reasoning

The only thing more fun than having NO reason for doing something is having a STUPID reason for doing something.

Do Left-Handed People Get Sick Less Often?

I'm left-handed, and historically I haven't been sick very often.  I had a silly thought that it's perhaps because I always shake hands with my right hand when I encounter people, and don't use my right hand for much anything else.  So with potential germs from others primarily on my non-dominant hand, maybe that's a reason I don't get sick as much.

I'm guessing nobody has tested that through a controlled science experiment yet, but I have no idea.

Effective Lies

 Lies can be very effective when they make more sense than the truth.