Simultaneous Equations

I am writing this post because I've seen several problems with the solving of simultaneous equations.
Eg. If xy = 6 and x+y = 5 find x and y
      x = 5-y
      5y-y^2 = 6
      (y-2)(y-3)=0
      y= 2 or 3
If y = 2, x=3 and if y = 3, x =2

If you are trying to solve a simultaneous equation with a function and a non-function, then final substitution must be made into the function otherwise it may yield incorrect results.
Eg. Solve simultaneously
      x^2 + y^2 = 16 (circle graph - not a function)
      3x - 4y - 20 = 0
      y = (3x-20)/4
Substituting into the first equation gives you
x^2 + [(3x-20)/4]^2 = 16
x^2 + (9x^2 - 120x + 400)/16 = 16
16x^2 + 9x^2 - 120x + 400 = 256
25x^2 - 120x + 144 = 0
(5x - 12)^2 = 0
x = 12/5 At this point, x must be substituted into the function to find the y value, otherwise it will give more than one result because there are two y values for just about every x value.

Another note sqrt(x^2 + y^2) ≠ x+y
(x + y)^2 ≠ x^2 + y^2