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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