Unknown

4.8. Problems on Systems of Equations 151

4.8 Problems on Systems of Equations

Solve the following systems of equations:

1. x4 + y2 - xys - 9x/8 = 0
   y4 + x2 - yx3 - 9y/8 = 0 (for real x, y).


2. x/a + b/y + c/z = a/x + y/b + c/z = a/x + b/y + z/c = 1.


3. x+y+r=o
   x2 + y2 + z2 = 6ab
   x3+y3+z3=3(a3+b3).


4. x+y--z=2
   x2 + y2 - z2 = 8 - 2xy
   x3 + y3 - z3 = 86 - 3XY%.


5. xy + yz + zx = a2 - x2 = b2 - y2 = c2 - z2.


6. 31x2y2 - 7y4 - 1121~ + 64 = 0
   x2-7xy+4y2+8=0.


7. x2 - (y - .z)~ = a2
   y2 - (z - x)” = b2
   r2 - (x - y)2 = c2
   in which a, b, c are constants different from 0.


8. Let a, b, c be different real numbers. Show that the only real solution
   of the system of equations

x+y+z=o

ax+by+cz=O

x3 + y” + z3 = 3(b - C)(C - a)(a - b)

isx=b-c,y =c-a,z=a-b.

9. x(y + z) = a
   y(% +x) = b
   z(x + y) = c
   where it is understood that the greatest of a, b, c is less than the sum
   of the other two.


10. (a) U+ 21 = (l/u) + w = (l/v) + (l/w)
    (b) u + 2, + (11~) = (l/u) + w + (U/W) = (l/v) + (l/w) + VW =
    uzJ+ (w/u) + (l/VW).


11. x2+y2 = 13 x3 + y3 = 35.