Combine the second two equations to eliminate the Y's. Then solve one of the variables in the new equation (say X) in terms of the other (Z). Then multiply and combine two of the original equations to get the X's gone. Then solve for Y in terms of Z. Then substitute the values in for X and Y (the ones in terms of Z) so that the entire equation is all Z's. Then solve and substitute the Z value into the two-variable equations to solve for X and Y. Then check to make sure that it works.
(If it doesn't work, try getting rid of the same variable both times you combine equations. Then multiply and combine the two new equations to eliminate one variable to find the value of the other. Then substitute the value again, and check.)