+67 In this problem, we will consider this system of simultaneous equations:
\(\begin{array}{r@{~}c@{~}l l} 3x+5y-6z &=&2, & \textrm{(i)} \\ 5xy-10yz-6xz &=& -41, & \textrm{(ii)} \\ xyz&=&6. & \textrm{(iii)} \end{array}\)
Let a=3x, b=5y, and c=-6z.
Given that (x,y,z) is a solution to the original system of equations, determine all distinct possible values of x+y.
+118677 I have forgotten how to do this. 
I would like to be reminded of the method though. Any takers?
+118677 Thanks very much Alan,
I finally managed it with a different method.
\(\begin{array}{r@{~}c@{~}l l} 3x+5y-6z &=&2, & \textrm{(i)} \\ 5xy-10yz-6xz &=& -41, & \textrm{(ii)} \\ xyz&=&6. & \textrm{(iii)} \end{array} \)
Let
\(a=3x \qquad\rightarrow \qquad x=\frac{a}{3}\\ b=5y \qquad\rightarrow \qquad y=\frac{b}{5}\\ c=-6z \qquad\rightarrow \qquad z=\frac{c}{-6}\\\)
The above equations become
\(a+b+c=2\\ ab+bc+ac=-123\\ abc=-540\)
This gave me a hint:
If a,b and c are the roos of a monic cubic equations then we have.
Note: I used g because x has a different meaning in this question.
\(g^3-(a+b+c)g^2+(ab+bc+ac)g-(abcd)=0\\ g^3-(2)g^2+(-123)g-(-540)=0\\ g^3-2g^2-123g+540=0\\ \)
I used the wolfram|alpha calculator to factoize this and got
\((g-9)(g-5)(g+12)=0\)
so a,b,c = -12,5,9 (any order)
These can be arranged in 3! = 6 ways so there will be 6 answers.
To show all the possible answers I made up a Excel table:
Alan thinks that there are 9 solutions but i don't know what the others could be ....
Notice that my last solution is the one that Alan found.
