6(x+y) = 5xy      ................1
21(y+z) = 10yz        ...........2
14(z+x) = 9zx   .........3
 
eq1 = 6(x+y) = 5xy
       1/x + 1/y = 5/6      ...............4
 
eq2 = 21(y+z) = 10yz 
             1/y + 1/z = 10/21       ................5
 
eq3 = 14(z+x) = 9zx
            1/z+1/x = 9/14      ...............6
 
eq 5 + eq 6 - eq 4
 2/z = 10/21 + 9/14 - 5/6
  z = 7
 x = 2
 y = 3
 
						

							
6(x+y)=5xy 
21(y+z)=10yz 
14(z+x)=9zx
6x + 6y = 5xy ----------------------------------(i)

21y + 21z = 10yz --------------------------------------(ii)

14z + 14x = 9zx --------------------------------------------------(iii)
dividing (i) by xy we get
1/y + 1/x = 5/6 -------------------------------(iv)
dividing (ii) by yz we get
1/z + 1/y = 10/21 ----------------------------------(v)
dividing (iii) by zx we get
1/x + 1/z = 9/14 --------------------------------------(vi)


subtracting (vi) from (v)
1/y - 1/x = 10/21 - 9/14 = -1/6 ---------------------- (vii)
Adding (iv) and (vii)
2/y = 5/6 - 1/6 = 4/6 = 2/3
1/y = 1/3
==> y=3 -----------------------------------------------(A)
substituting the value of (A) in (v) we get :

1/z = 10/21 - 1/3 =1/7
==> z=7 -----------------------------------------------(B)
substituting the value of (B) in (vi) we get :
1/x = 9/14 - 1/7 = 1/2
==> x = 2 ------------------------------------------------(C)
so the answers are
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x = 2 ----------------- from (C)   |

y = 3 ----------------- from (A)   |

z = 7 ----------------- from (B)   |

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