Solve this system of the four equations of four unknowns $a, b, c, d>0 $
$$ 165(a+b+c)=abc\tag1 $$ $$220(a+b+d)=abd \tag2 $$ $$297(a+c+d)=acd\tag3 $$ $$540(b+c+d)=bcd \tag4 $$
I tried to solve it by following method. On substituting the values of two variables c & d from equations (1) & (2) in the equations (3) & (4), I got
$$297a^3b^2-36300a^3-72600a^2b+78045ab^2-10781100a-21562200b=0\tag5 $$
$$540a^2b^3-36300b^3-72600ab^2+171600a^2b-39204000a-19602000b=0\tag6 $$
How to proceed? Any help is greatly appreciated.
When you have two non-linear equations in two unknowns,
$$F_1(x,y)=0$$
$$F_2(x,y)=0$$
you can use resultants to eliminate one unknown. There is no point in doing it by hand since nowadays, one can use computer algebra systems to do this efficiently. The command in Mathematica is,
$$\text{Resultant}[F_1(x,y),F_2(x,y),y]$$
where $y$ is the variable to be eliminated. Eliminating your $b$ using WolframAlpha, one finds,
$$a (-165 + a^2) (165 + a^2) (220 + a^2) (88944075 + 1281390 a^2 + 5962 a^4 + 9 a^6)=0$$