Four Unknowns
Solve this system of equations:
x + y + z + w = 10
x^2 + y^2 + z^2 + w^2 = 30
x^3 + y^3 + z^3 + w^3 = 100
xyzw = 24
(brute force acceptable ... however, as Chris would say, "the method is much more interesting")
x + y + z + w = 10
x^2 + y^2 + z^2 + w^2 = 30
x^3 + y^3 + z^3 + w^3 = 100
xyzw = 24
(brute force acceptable ... however, as Chris would say, "the method is much more interesting")
posted by Zaux at
5:44 AM
13 Comments:
wonder why the blogger dashboard will not allow superscripts or subscripts?
Is the last equation supposed to b e xyzw=24?
yes anonymous ... thanks, I will correct
corrected ... :-)
Here's an interesting method: type the first two equations, with the word AND between them, into wolfram alpha. It gives you an integer solution, which you can then check in the last two equations. :-) :-)
It even works with three simultaneous equations!
Interesting. You'd think that the symmetry would => w = x = y = z
Using w+x+y+z = 10 => w = x = y = z = 2.5.
But that doesn't work. I'll be back.
There are 24 solution sets. Go to www.wolframalpha.com and paste
the following into the equation editor:
w+x+y+z=10,w^2+x^2+y^2+z^2=30,w^3+x^3+y^3+z^3=100,w x y z=40
Or just paste the following into your browser's address bar.
http://www.wolframalpha.com/input/?i=w%2Bx%2By%2Bz%3D10%2Cw%5E2%2Bx%5E2%2By%5E2%2Bz%5E2%3D30%2Cw%5E3%2Bx%5E3%2By%5E3%2Bz%5E3%3D100%2Cw+x+y+z%3D40.
There is no way that I'm going to do that by hand. Unless you're
very lucky, cubics and up are too painful. In general, quintics
and up cannot be except numerically.
By inspection, 1,2,3,and 4 is a solution of the first and fourth equation and satisfies the second and third. Since the equations are symmetrical in z,y,z,and w, the other 23 permutations of 1,2,3, and 4 are also solutions. These are all the solutions,since the product of the degrees of the equation is 4!
Oooops. Is see that I used wxyz = 40 instead of 24.
Here's a fixed up link:
http://www.wolframalpha.com/input/?i=w%2Bx%2By%2Bz%3D10%2Cw%5E2%2Bx%5E2%2By%5E2%2Bz%5E2%3D30%2Cw%5E3%2Bx%5E3%2By%5E3%2Bz%5E3%3D100%2Cw+x+y+z%3D24
If you check the erroneous one, tou'll see why I didn't want to try it.
Hindsight is 20-20. Just knowing the problem is Diophantine and w,x,y,z are different is enough to permit a quick solution. As that couldn't be guaranteed, it would have required quite a lot of tedious effort to get the result.
the answer is
w=1
X=2
y=3
z=4
that may be the answer but can anyone show how to get the answer without using a solver? im trying but i can only get variables existing over intervals, not an actual step by step process to arive at the answer.
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