A cubic system of equations

This video is about a cubic system of equations
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[angelmendez-rivera]

For anyone who may be confused by the method in the video, asking "why can we assume that y and x are proportional?" The answer is that, technically, there is no assumption built into the substitution. All he did was perform an arbitrary change of variables. If it helps, you can motivate this change of variables by realizing that every nonzero real number is equal to the product of two nonzero real numbers in uncountably infinite many ways. Perhaps it will all be clearer if instead of using the letter m, which is suggestive, the letter z is used.

[notlin]

Both methods are very interesting and are two excellent strategies for a lot of questions with 2 equations and two variables.

[hkemal]

Why was the equal sign so humble?

Because she knew she wasn’t greater than or less than anyone else.

[wes]

Excellent. Try with this. A sphere has volume V=(4/3)pi(2+sqrt(5)) = (4/3)pi*r^3. Let r = x + sqrt(5)y so r^3 = 2+sqrt(5) = (x + sqrt(5)y)^3. This gives two cubic equations in x and y with one solution x = y = 1/2 and r = (1 + sqrt(5))/2.

[jtom]

much faster is to use Horner's method to try if a number is the root. Immediately u also obtain the result of polynomial division

[joaquingutierrez]

Nice problem !! I was not able to solve the equation, I tried to get a perfect cube but I was not able to do so.

Anyway, I liked how you solved the problem. Thanks for the video :)

[GourangaPL]

you got 3 values of y in terms of x, wouldn't solving 2nd equation as a quadratic in y give only 2 of them?

[elkincampos]

There is 9 points or infinity by Bèzout theorem. We should see to solution in the complex numbers.

[SyberMath]

I like this problem. I present two solutions here. Which method do you think is better?

[neuralwarp]

But what if y ≠ mx ?
Are you excluding possible solutions? Proof?

[arjunverma]

could u explain the validity of assuming a linear relation between x and y(x=my), as far as i believe there doesnt have to be a relation between x and y in the first place cuz they both are independent in their respective aspect

[michaelempeigne]

from subtracting the equations i get : 2x^3 + 5x^2 * y + y^2 * ( y - x ) = 7
then it is pretty obvious that y = 1 and x = 1 is a solution to the system.

[holyshit]

Using substitutions i got cubic with one integer root easy to guess
Substitution method may lead us to extraneous solutions

[moros_gamer]

Thank you!, but could you help me with this?
Solve for positive integers x, y
1+22x+20y=xy

[attila]

Cool but i liked diophantine eqns more

[mali-bqiv]

Hocam emeğinize ve zihninize sağlık. 👏👏👏