Can someone explain this answer to me please?

The binary operation * on the set {x,y,z} is associative.

You are given part of the operation table for * below. Fill in the gaps. There is only one way to fill in the gaps for an associative operation.

And its a table that has x, y, and z on the x and y axis. It gave three values to us already. x * y = y, y * z = z and z * x = x. By the way, when I say x * y, I mean x is on the column side and y is on the row. So x on the left side and y on the top.

Thank you very much.

As an example, you have
x*y = y
y*z = z
but you dont know x*z = ... so can you think how you can get it by "multiplying" the first equation by ... and subbing the second.

As an example, you have
x*y = y
y*z = z
but you dont know x*z = ... so can you think how you can get it by "multiplying" the first equation by ... and subbing the second.

Thank you very much that was very helpful. I have another question. Lets say that in a row (horizontal) you have, for example, x and z already filled in. For an associative operation table, is it safe to assume that the blank space in that row is y? And this applies for other examples aswell, eg; y and z, so you can assume that the blank space is x?