Question

Selesaikan dari sistem persamaan
1/x + 1/y + 1/z = 5
2/x - 3/y - 4/z = -11
3/x + 2/y - 1/z = 6


tentukan hasil x+y+z

ket: / (per) contoh: 1 per x.

tolong yahhh

Answer 1

Misalkan : 1/x = p
                   1/y = q
                   1/z = r 
maka :
p+q+r = 5 ...............(1)
2p-3q-4r = -11........(2)
3p+2q-r = 6............(3)

eliminasi ...(1) & ...(3)
p+q+r = 5
3p+2q-r = 6
------------------- +
4p + 3q = 11 ........(i)

eliminasi ...(1) & ...(2) dengan syarat persamaan (1) dikalikan 4, jadi :
4p+4q+4r = 20
2p-3q-4r = -11
-------------------- +
6p + q = 9 ........(ii)

eliminasi ....(i) & (ii) dengan syarat persamaan (ii) dikalikan 3 agar diperoleh nilai p , jadi :
4p+3q = 11
18p + 3q = 27
-------------------- - 
-14p = -16
14p = 16
p = 16/14
p = 8/7
-> 4p + 3q = 11
    4(8/7) + 3q = 11
    (32/7) + 3q = 11
    3q = 11 - (32/7)
    3q = (77/7) - (32/7)
    3q = 45/7
    q = (45/7)/3
   q = 45/21
    q = 15/7
-> subs. nilai p dan q ke persamaan (1)
  p + q + r = 5
(8/7) + (15/7) + r = 5
(23/7) + r = 5 
r = 5 - (23/7)
r = (35/7) - (23/7)
r = 12/7

---> p = 8/7
       q = 15/7
       r = 12/7
p = 1/x => x = 1/p= 1/(8/7) = 7/8
q = 1/y => y = 1/q = 1/(15/7) = 7/15
r = 1/z => z = 1/r = 1/(12/7) = 7/12

jadi, x+y+z = 7/8 + 7/15 + 7/12 
                   = (105+56+70)/120
                   = 231/120