x^2 + y^2 + z^2 = 18
xy + xz + yz = -7
Let x, y, and z be integers satisfying the system of equations above. Find xyz.
+874 $(x+y+z)^2 - 2(xy + xz + yz) = x^2 + y^2 + z^2$
$(x+y+z)^2 - 2(-7) = 18$
$(x+y+z)^2 = 4$
$x+y+z = 2$
$x^2 y^2 + 2 x^2 y z + x^2 z^2 + 2 x y^2 z + 2 x y z^2 + y^2 z^2 = 49$
$x^2 y^2 + x^2 z^2 + y^2 z^2 + 2xyz(x + y + z) = 49$
$\frac{xyz^2}{z^2} + \frac{xyz^2}{y^2} + \frac{xyz^2}{x^2} + 4xyz = 49$
$xyz \left(\frac{xyz}{z^2} + \frac{xyz}{y^2} + \frac{xyz}{x^2} \right) + 4xyz = 49$
$xyz \left(\frac{xyz(x^2 y^2 + y^2 z^2 +x^2 z^2)}{xyz^2} \right) + 4xyz = 49$
$xyz \left(\frac{18}{xyz} \right) + 4xyz = 49$
$18 + 4xyz = 49$
$4xyz = 31$
$xyz = \boxed{\frac{31}{4}}$
+874 I made a mistake with the $\frac{18}{xyz}$, though I am scared to edit the latex : Do you think you can manipulate $x^2 y^2 + y^2 z^2 + x^2 z^2$ into something you already know? The answer follows from there.
