Triples Relationship

Algebra Level 4

x = y z y = x z z = x y \large {{ x=yz } \\ {y=xz } \\ {z=xy } }

Let the triples ( x , y , z ) (x,y,z) of non-zero real numbers. How many ordered triplets ( x , y , z ) (x,y,z) satisfy the system of equations above?


The answer is 4.

Abdeslem Smahi by Abdeslem Smahi , 24, Algeria

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2 solutions

Ravi Dwivedi
Jul 14, 2015

x y z = ( x y z ) 2 xyz=(xyz)^2 ⟹ x y z = 1 \implies xyz=1 or 0 0

But we need to find non zero solutions ⟹ ( x , y , z ) = ( 1 , 1 , 1 ) , ( − 1 , − 1 , 1 ) , ( 1 , − 1 , − 1 ) , ( − 1 , 1 , − 1 ) \implies (x,y,z)=(1,1,1),(-1,-1,1),(1,-1,-1),(- 1,1,-1)

So answer is 4 \boxed{4}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Bonus question : Can you generalize this? That is, If a 1 , a 2 , … , a n a_1, a_2, \ldots, a_n are non-zero numbers such that any a i a_i is the product the other a i a_i 's. What is the ordered solutions for ( a 1 , a 2 , … , a n ) (a_1,a_2,\ldots,a_n) for n ≥ 3 n\geq 3 ? In this case, n = 3 n=3 yield 4 solutions.

(Updated)

Now it is allright

Ravi Dwivedi - 5 years, 11 months ago
Abdeslem Smahi
Jul 12, 2015

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

You made a few careless mistakes.

There's a much simpler approach.

Hint : Multiply them all together.

The last 2 answers are incorrect since they do not meet the 3 relations coz we get 1×1 which is never equal to -1. So first 2 answers are correct but the last 2 answers needs to be seen through.

YoYo Pro - 5 years, 11 months ago

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yeah u're right i just made a mistake

the triples are ( 1 , 1 , 1 ) ; ( − 1 , 1 , − 1 ) , ( − 1 , − 1 , 1 ) ; ( 1 , − 1 , − 1 ) (1,1,1);(-1,1,-1),(-1,-1,1);(1,-1,-1)

sorry for the mistake , thank you for your observation

Abdeslem Smahi - 5 years, 11 months ago

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Yeah this is absolutely correct

YoYo Pro - 5 years, 11 months ago

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