Given x, y, and zinRR^+, what is the solution to the system of equations xy=a, yz=b, and xz=c?

1 Answer
Dec 28, 2015

{(x = sqrt((ac)/b)), (y = sqrt((ab)/c)), (z = sqrt((bc)/a)):}

Explanation:

Before continuing, let's note that as x, y, z > 0 we have a, b, c >0 and so we may take square roots of any product or quotient thereof, and do so without considering negative solutions.

{(xy = a),(yz = b),(xz=c):}

From the first equation, we have

y = a/x

Substituting this into the second equation:

b = yz = a/xz

=> z = b/ax

Substituting this into the third equation:

c = xz = b/ax^2

=> x^2 = (ac)/b

=> x = sqrt((ac)/b)

Substituting our solution for x into our intermediate result from working on the first equation:

y =a/x = a/sqrt((ac)/b) = sqrt((ab)/c)

Substituting our solution for x into our intermediate result from working on the second equation:

z = b/ax = b/asqrt((ac)/b) = sqrt((bc)/a)

:.{(x = sqrt((ac)/b)), (y = sqrt((ab)/c)), (z = sqrt((bc)/a)):}