+223 Given the system of equations \(\begin{align*} xy &= 6 - 2x - 3y,\\ yz &= 6 - 4y - 2z,\\ xz &= 30 - 4x - 3z, \end{align*}\)find the positive solution of x.
+129852 xy = 6 - 2x - 3y xz = 30 - 4x - 3z
xy + 3y = 6 - 2x xz + 3z = 30 - 4x
y ( x + 3) = 6 - 2x z( x + 3) = 30 - 4x
y = (6 - 2x) / ( x + 3) z = ( 30 - 4x) / ( x + 3)
yz = 6 - 4y - 2z
(6 - 2x) /( x + 3) * ( 30 - 4x)/ (x + 3) = 6 - 4 (6 - 2x)/(x + 3 ) - 2 ( 30-4x)/ ( x + 3)
Multiply through by (x + 3)^2
(6- 2x) * (30 - 4x) = 6 ( x + 3)^2 - 4 (6-2x)(x + 3) - 2(30-4x)(x + 3)
180 - 84x + 8x^2 = 6(x^2 + 6x + 9) - 4(6x - 2x^2 + 18 - 6x) - 2 ( 30x - 4x^2 + 90 - 12x)
180 - 84x + 8x^2 = 6x^2 + 36x + 54 - 24x + 8x^2 - 72 + 24x - 60x + 8x^2 - 180 + 24x
Simplify
180 - 84x + 8x^2 = 22x^2 -198
14x^2 + 84x - 378 = 0 divide through by 14
x^2 + 6x - 27 = 0 factor
(x - 3) ( x + 9) = 0
The positive solution is x -3 = 0 ⇔ x = 3
