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given positive integers x y and z 

they satisfy the following equations:

7x^2 - 3y^2 + 4z^2 = 8

16x^2 -7y^2 + 9z^2 = -3

 

what's the value of x^2 + y^2 + z^2 ?

 Nov 24, 2018
 #1
avatar+107348 
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7x^2 - 3y^2 + 4z^2 = 8            (1)

16x^2 -7y^2 + 9z^2 = -3          (2)

 

Multiply the first equation through  by 7   and the second through by -3

 

49x^2 - 21y^2 + 28z^2 = 56

-48x^2 + 21y^2 - 27z^2 = 9        add these

 

x^2   +  z^2 = 65

 

Possible integer values for    (x , z) are ( 4, 7)   or  ( 7, 4)

 

Since x^2, z^2  are arbitrary....we can sub (4, 7)  into (1) for (x, z)

And we have

 

7^3 - 3y^2 + 4^3 = 8

 

-3y^3 + 407 = - 3

 

-3y^2 =  - 410

 

This does not produce an integer for y

 

Sub (7,4) = (x, y)  into (1)

 

7(4)^2 - 3y^2 + 4(7)^2 = 8

 

112 - 3y^2 + 196 = 8

 

-3y^2 =  -300

 

y^2 = 100

 

y = 10

 

 

So

 

x^2 + y^2 + z^2   =

 

65 + 100   =

 

165

 

 

cool cool cool

 Nov 24, 2018

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