+160 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 ?
+128407 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
