Repeated... again.

If we label the equations 1, 2, and 3 from top to bottom, then

2-1: ${ z }^{ 2 }-{ x }^{ 2 }=9$

3-2: ${ x }^{ 2 }-{ y }^{ 2 }=7$

3-1: ${ z }^{ 2 }-{ y }^{ 2 }=16$

Hold on... the first and last equations look like a pythagorean triple I know. Let's bash those numbers in and see what we get. If z=5, x=4, and y=3, then

2-1: ${ 5 }^{ 2 }-{ 4 }^{ 2 }=9$

3-2: ${ 4}^{ 2 }-{ 3 }^{ 2 }=7$

3-1: ${ 5 }^{ 2 }-{ y }^{ 3 }=16$

All these statements are true, so (x,y,z) = (4,3,5) is a solution, and therefore the solution is $3\times 4\times 5=60$

(I am aware that this solution lacks the rigour I would like, but the solution jumped so clearly out at me, I could not help but see it)