max f(x,y,z)=5xy+5xz+3yz subject to 2xyz=1920

With regards to the above question relating to lagrange multipliers I understand I have to differentiate with regards to x,y and z. But I do not understand how:

Using pair-wise divisions we get:

(5x + 8z)/( 5x + 3z) = y/x firstly, is there an error? is the 5x meant to be 5y?

( 5x + 3z)/(8x + 3y) = z/y

(5y + 8z)/(8x + 3y) = z/x

secondly, could you explain in detail how we simultaneously solve each pair please. i.e. when we differentiate we should get 3 different equations, but how do we get to the different values x,y and z?

In other words, how do we solve those equations simultaneous to obatin the values of x,y and z? Can you provide a step by step explanation as this is where I am getting stuck.