Symmetric System of Equations x 2 </msup> &#x2212;<!-- − --> y 2

Josie Sparks

Josie Sparks

Answered question

2022-06-02

Symmetric System of Equations
x 2 y 2 = z
y 2 z 2 = x
z 2 x 2 = y

Answer & Explanation

Angel Chan

Angel Chan

Beginner2022-06-03Added 6 answers

Add x 2 y 2 = z and y 2 z 2 = x to get x 2 z 2 = x + z. This means either x + z = 0, or x z = 1. Similarly, either y + x = 0 or y x = 1, and either z + y = 0 or z y = 1.
Now there are a total of 8 different combinations of equations to deal with.
1. x + z = 0 , y + x = 0 , z + y = 0
2. x + z = 0 , y + x = 0 , z y = 1
3. x + z = 0 , y x = 1 , z + y = 0
4. x + z = 0 , y x = 1 , z y = 1
5. x z = 1 , y + x = 0 , z + y = 0
6. x z = 1 , y + x = 0 , z y = 1
7. x z = 1 , y x = 1 , z + y = 0
8. x z = 1 , y x = 1 , z y = 1
And then you just proceed to solve for x, y, and z, and find which cases make sense and which ones don't.

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