Assuming x, y, and z are positive real numbers satisfying:
xy - z = 15
xz - y = 15
yz - x = 15
then, what is the value of xyz?
\(\phantom{yz - x = 0}\)
\(\phantom{xz - y = 0}\)
Hi Guest!
Rewrite the first two equations as follows:
\(xy=15+z\) (1)
\(xz=15+y \) (2)
Subtracting (2) from (1) gives:
\(x(y-z)=-(y-z)\) (Notice: \(z-y=-(y-z)\) ).
\(x(y-z)+(y-z)=0\) (Add (y-z) to both sides to make R.H.S = 0 , and factor (y-z)):
\((y-z)(x+1)=0\)
Thus, \(y=z, \text{or } x=-1 (rejected)\) (As given x,y,z > 0).
Therefore, \(y=z\) only.
Also, using the second and third equation:
\(xz=15+y\) (2)
\(yz=15+x\) (3)
Subtracting (3) from (2) to get:
\(z(x-y)=-(x-y)\)
\(z(x-y)+(x-y)=0\)
\((x-y)(z+1)=0\)
Therefore, \(x=y\) (z>0 so z=-1 is rejected).
But we found that: \(y=z\)
Thus, It must mean that they are all equal to each other. That is, \(x=y=z\)
So our system of equations becomes:
\(x^2-x=15\) \(\iff x^2-x-15=0 \iff x=\frac{1+\sqrt{61}}{2}\) (A quadratic equation in x, which can be solved by the quadratic formula.)
So the desired answer is:
\(xyz=x^3=\frac{(1+\sqrt{61})^3}{8}=23+8\sqrt{61}\)
I hope this helps!