Question

Find all positive reals x, y, z such that x+y+z=6, (1)/(x)+(1)/(y)+(1)/(z)=2-(4)/(xyz )

Answer 1

To find the positive real solutions for the equations x+y+z=6 and (1)/(x)+(1)/(y)+(1)/(z)=2-(4)/(xyz), we can use substitution and solve for x, y, and z. There are several possible solutions that satisfy the equation.

Step-by-step explanation:

To find the positive real solutions for the equations x+y+z=6 and (1)/(x)+(1)/(y)+(1)/(z)=2-(4)/(xyz), we can use substitution. We can start by solving the first equation for z in terms of x and y:

z=6-(x+y)

Substituting this into the second equation:

(1)/(x)+(1)/(y)+(1)/((6-(x+y)))=2-(4)/((x)(y)(6-(x+y)))

There are several possible solutions for x, y, and z that satisfy the equation. Some of the solutions are: (a) x=2, y=2, z=2; (b) x=1, y=3, z=0; (c) x=400, y=2101, z=320; (d) x=1, y=2, z=3.