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The system of equations
\frac{xy}{x + y} = 0, \quad \frac{xz}{x + z} = 0, \quad \frac{yz}{y + z} = 1
has exactly one solution.  What is $z$ in this solution?

 Jan 14, 2024
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\(\frac{xy}{x + y} = 0, \quad \frac{xz}{x + z} = 0, \quad \frac{yz}{y + z} = 1 \)

 

Either  x  =  0      or    y, z  = 0

But the second cannot be true because it would make the  last fraction undefined

So  x =  0

 

And  (assuming integer solutions for y, z )

 

(yz)  /  (y + z)  =  1

 

yz  =  y + z 

 

zy  - z  =  y

 

z ( y - 1)  =  y

 

z  =  y / (y -1)

 

y       z

2      2

 

z =  2

 

cool cool cool

 Jan 14, 2024
edited by CPhill  Jan 14, 2024

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