That's a good start. All you have to do now is find the express the value of two variables in terms of one common one.
I chose to solve that problem by expressing the value of x in terms of y, and the value of z in terms of y.
The equation that has x and y in it is xy = 24. I divided both sides of that equation by y to get x on one side, so the equation now states:
x = 24/y
The equation that has y and z in it is yz = 3. I divided both sides of that equation by y to get z on one side, so the equation now states:
z = 3/y
Then the third equation you came up with, xz = 8, comes into play. x = 24/y and z = 3/y, so I can substitute 24/y for x in the equation: xz = 8. I can also substitute 3/y for z in that same equation. So, xz = 8 becomes (24/y)(3/y) = 8.
Then you multiply the numerators and the denominators so your equation then reads, 72 over the y squared = 8.
You want to find the value of y so you multiply both sides of that equation by y squared. Then you get 72 = 8 times y squared.
Next, you divide both sides of the equation by 8, and so you get 9 = y squared. Then you find the square root of each side, leaving you with y = 3.
With that said, you can plug in 3 for the y in the other two equations, yz = 3 and xy = 24, to get the values of z and x.
If you do that, then you'll find that z = 1 and x = 8. Hope that helps