This Video Will Make You Better At Algebra

How do you suppose you might solve this interesting equation x^y=y^x ? There are more than a few ways to do this! We will do it by introducing a parameter and solving for x and y in terms of it.

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

x ^ y = y ^ x

#math #brithemathguy #algebra

[BriTheMathGuy]

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[axbs]

I always find it so weird how you can add more variables to make a problem less confusing

Thank you for this insightful video and I’m really enjoying this series!

[dukenukem]

This is one of the problems that I did with my son when I introduced him to the Lambert W function. I’ll have to add this alternative solution technique to that worksheet packet. Thanks for sharing!

[zone]

legit took me 40 seconds to get even more confused than i am during class

[berengriffiths]

What's even more interesting is that this result still holds for irrational AND complex x and y (although solving for m to find y from a given x involves the use of the lambert w function). It's fascinating that a seemingly specific case at the beginning leads to *all* solutions! Great video!

[InstigationMex]

Quickly becoming one of my favorite math channels. Such clear and intuitive explanations

[michiell.]

You didn't include the solutions where y=x (when m=1, all pairs (x, x) are solutions except for x=0). Other than that, very interesting.

[tausifraza]

X = 2 & Y = 4 is also satisfies the equation
Other possible combinations
X = -2 & Y = -4
X = -2 & Y = 4 (and vice versa)

[naivedyam]

Hey Bri. Here's a problem created by me. Do give it a try-

If (dy/dx)^2 + y^2 = 4y, find derivative of dy/dx at y = 2 - root2. Condition is you can't use differentiation at any step.

[daniel-kun]

at 0:40 to assume y=mx you will need v|u but nothing states that. Is the whole thing wrong?

[nice_mf_ngl]

I'm currently in 7th but i am preparing for the maths Olympiad in india (IOQM and INMO), your videos are super relevant and interesting, thanks for this video and please try INMO questions they are really interesting

[yj_holic]

Never thought about it in terms of fractions! I only new 2^4= 4^2 = 16

[jasnoor-d-]

Hey which editing app do you use ? I would really appreciate if you tell

[MessiBoT]

2:48 The most important sentence for me, I really needed it.

[manucitomx]

I love these very clear videos.
Thank you for your hard work. (I do miss the ones where we actually saw you, though).

[nicolastorres]

2:20 we can actually graph the solution curve r(m) = (x(m), y(m)) in geogebra

[batteryjuicy]

So m is he patameter of the second line that intersects the y^x =x^y line. So if we subtract the 2 equation you found the end, we get the equation of the second line. Did i understand it correctly?

[foji-video]

you actually get to that solution without involving going through the "let's consider this as if it was a natural exponent" part, with may seem non-rigorous.
The reasoning holds by just going with factorization:
x^y = y^x --> (x^(y-1)) x = (y^(x-1)) y . Now you can call x^(y-1) as your u and y^(x-1) as your v, and you get that m.
Of course m depends on x and y, but as you've shown you can actually get the local "inverse" of the m(x, y) function and get y(m) and x(m)

[aren]

I love complex problems and i love it even more when i can solve them with some help

[chunqiangli]

How can you be sure that X is linear with Y？
You can not take the relation of linearity for granted