Show that the system of equations { <mtable columnalign="left left" rowspacing=".2em" c

Kaeden Woodard

Kaeden Woodard

Answered question

2022-05-21

Show that the system of equations
{ x 4 + y 4 + 2 z 4 = 1 x 2 + y 2 + 2 z 2 = 7
has no real solutions.

Answer & Explanation

Rubi Boyle

Rubi Boyle

Beginner2022-05-22Added 14 answers

Write a = x 2 , b = y 2 , c = z 2 so we have
a 2 + b 2 + c 2 = 1 a + b + 2 c = 7
The first equation is the unit sphere and the second is a plane; a real solution corresponds to an intersection, where the origin-plane distance is 1. But the shortest distance from a + b + 2 c = 7 to the origin is 7 6 > 1. Thus the plane does not intersect the sphere and the system in a , b , c has no real solution, implying that the original system also has no real solutions.

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