# Solve for 3 variables with 2 equations

1. May 2, 2012

### staples

1. The problem statement, all variables and given/known data
Solve the system of equations for x, y, and z.
-------------------------
$$4x^4+8y^4+z^4=9$$
$$x^8+y^8+z^8=1$$
--------------------------

2. Relevant equations
3. The attempt at a solution
Both equations can be rearranged and factored (difference of squares). I've tried it and it doesn't seem to help solve the problem. Any hints? Thanks :)

Last edited: May 2, 2012
2. May 2, 2012

### tiny-tim

welcome to pf!

hi staples! welcome to pf!

(i assume that's x8, not xy?)

have you tried the obvious substitution?

3. May 2, 2012

### Staff: Mentor

Is the exponent on x in the 2nd equation a typo? Just checking.

4. May 2, 2012

### staples

Sorry about that. That was a typo. It's corrected now.
tiny-tim: I don't see it

5. May 2, 2012

u = x4 … ?

6. May 2, 2012

### Staff: Mentor

And similar substitutions for y4 and z4.

7. May 2, 2012

### staples

So I substituted x^4 = u, y^4 = v, z^4 = w.
I still don't see how it would help. The only thing I could see to isolate one variable in the first equation and substitute that in to the second equation. That still leaves us with 2 variables. I don't understand how we can solve for 3 variables with 2 equations. I always thought if you have n variables you need at least n equations. Please explain. Thanks

8. May 2, 2012

### SammyS

Staff Emeritus
What does each of the resulting equations describe?

$4u+8v+w=9$

$u^2+v^2+w^2=1$

How would you describe their intersection, if they intersect?

9. May 2, 2012

### Staff: Mentor

If you have n variables you need at least n equations to get a unique solution (a single point). Since you have only two equations in three variables, there won't be a unique solution. This means that the solution set will be all the points along some curve, assuming the two equations represent surfaces that intersect.

10. May 2, 2012

### staples

So like Marks says, the intersection will be infinite over a certain interval(s)? I don't understand what you mean by what they represent? 3 dimensional planes/surfaces? :uhh: Thanks.

11. May 3, 2012

### Staff: Mentor

In (u, v, w) space, the two equations are surfaces that might or might not intersect. If they intersect, they will do so at an infinite number of points - all the points along the curve of intersection.

12. May 3, 2012

### staples

I think I get it now. Divide the first equation by 9 and compare it to the second equation. This shows that u,v,w must be equal to their coefficients because of the square in the second equation. So u=4/9, v=8/9, w=1/9. Now we take the (positive and negative) fourth root of these values. So there are 8 solutions (WolframAlpha: http://www.wolframalpha.com/input/?i=%284x^4%2B8y^4%2Bz^4%3D9%29+%28x^8%2By^8%2Bz^8%3D1%29)

Is this correct? Can I just "compare" the two equations like I did? Thanks.

P.S. Mark: I think you're under the impression that I'm doing higher level math courses. I am only in high school and we've never worked with 3d coordinate systems...

13. May 3, 2012

### tiny-tim

the first equation represents a plane, the second represents a sphere

(so what will be the shape of the points where they intersect?)

14. May 3, 2012

### HallsofIvy

Staff Emeritus
Normally, you cannot solve two equations for specific values of three unknown numbers but here you can. It is crucial to this problem that $8^2+ 4^2+ 1^2= 64+ 16+ 1= 81= 9^2$

15. May 3, 2012

### staples

I'm guessing the answer is a circle or ellipse depending on how they intersect. But I highly doubt it that the teacher wants an answer using the geometry of the equations. After all we've never learned the equation of a sphere. I'm pretty sure he's looking for an algebraic approach...

I'm going to look in to what you said HallsofIvy. So I guess, my "solution" is wrong? A less subtle hint would be great :) I appreciate everyone's effort to help me. Thanks.

Last edited: May 3, 2012
16. May 3, 2012

### tiny-tim

a plane always intersects a sphere in a circle
yes, but knowledge of the geometry points you towards an easy algebraic approach

try substituting a b c for u v w so that a = constant, a2 + b2 + c2 = 1

17. May 3, 2012

### staples

|a|,|b|,|c| ≤ 1 ?

18. May 3, 2012

### staples

19. May 3, 2012

### tiny-tim

on the intersection, yes

conversion from u v and w axes to a b and c axes is just a rotation of the axes

(eg if instead of the usual x and y axes you use p = (x+y)/√2 and q = (x-y)/√2, you've simply rotated the axes by 45°, and x2 + y2 = 1 becomes p2 + q2 = 1)

here's another geometrical snippet that may help …

4u + 8v + w = 9 is a plane whose normal is the direction (4,8,1) …

if we call the endpoints of the diameter of the sphere parallel to (4,8,1) the north and south poles, then the solutions (in u,v,w coordinates) are a circle of latitude on the sphere

20. May 3, 2012

### SammyS

Staff Emeritus
... unless the plane is tangent to the sphere.