+0  
 
0
41
1
avatar+65 

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 ?

hearts123  Nov 24, 2018
 #1
avatar+92429 
+1

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

CPhill  Nov 24, 2018

10 Online Users

avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.