You can use NSolve and approximate computing. In such case the code is:
c1 = c2 = c3 = l1 = l2 = l3 = 1.;
NSolve[{
ca + cb + cc == c1,
ca lc cc + ca lb cb + cb lc cc + cb la ca + cc la ca + cc lb cb ==
c1 l2 c2 + c1 l3 c3,
ca cb cc lb lc + ca cb cc la lc + ca cb cc la lb == c1 c2 c3 l2 l3,
ca la + cb lb + cc lc == c1 l1 + c1 l2 + c1 l3 + c2 l2 + c3 l3,
ca cb la lb + ca cc la lc + cb cc lb lc == c2 c3 l2 l3,
ca cb cc la lb lc == c1 c2 c3 l1 l2 l3}, {ca, cb, cc, la, lb, lc}]
{{ca -> 0.177312 + 0.16778 I, cb -> 0.645259 - 9.19986*10^-6 I,
cc -> 0.177341 - 0.167806 I, la -> 1.50331 + 1.10002 I,
lb -> 7.49442 - 0.0000188153 I,
lc -> 1.50317 - 1.09989 I}, {ca -> 0.177343 - 0.167799 I,
cb -> 0.177334 + 0.167793 I, cc -> 0.64527 + 7.69554*10^-7 I,
la -> 1.50309 - 1.09987 I, lb -> 1.50327 + 1.09995 I,
lc -> 7.49426 + 0.0000573922 I}, {ca -> 0.177339 + 0.167798 I,
cb -> 0.177357 - 0.167798 I, cc -> 0.645273 - 0.00004873 I,
la -> 1.50322 + 1.09995 I, lb -> 1.50296 - 1.09988 I,
lc -> 7.49425 + 0.00060717 I}, {ca -> 0.645294 - 0.0000468883 I,
cb -> 0.177355 - 0.16779 I, cc -> 0.177339 + 0.167809 I,
la -> 7.49423 + 0.000272327 I, lb -> 1.50303 - 1.09989 I,
lc -> 1.50316 + 1.09983 I}, {ca -> 0.645299 - 0.0000183038 I,
cb -> 0.17733 + 0.167802 I, cc -> 0.177352 - 0.167804 I,
la -> 7.49422 - 0.0000507398 I, lb -> 1.50317 + 1.09986 I,
lc -> 1.50307 - 1.09984 I}, {ca -> 0.177331 - 0.167788 I,
cb -> 0.645295 - 0.000013715 I, cc -> 0.177317 + 0.167793 I,
la -> 1.50324 - 1.09994 I, lb -> 7.49403 + 0.000151404 I,
lc -> 1.50336 + 1.0999 I}}
And the code with Manipulate:
Manipulate[NSolve[{
ca + cb + cc == c1,
ca lc cc + ca lb cb + cb lc cc + cb la ca + cc la ca + cc lb cb ==
c1 l2 c2 + c1 l3 c3,
ca cb cc lb lc + ca cb cc la lc + ca cb cc la lb == c1 c2 c3 l2 l3,
ca la + cb lb + cc lc == c1 l1 + c1 l2 + c1 l3 + c2 l2 + c3 l3,
ca cb la lb + ca cc la lc + cb cc lb lc == c2 c3 l2 l3,
ca cb cc la lb lc == c1 c2 c3 l1 l2 l3}, {ca, cb, cc, la, lb, lc}],
{{c1, 1., "\!\(\*SubscriptBox[\(c\), \(1\)]\)"}, -10, 10, 1,
Appearance -> "Labeled", ImageSize -> Tiny},
{{c2, 1., "\!\(\*SubscriptBox[\(c\), \(2\)]\)"}, -10, 10, 1,
Appearance -> "Labeled", ImageSize -> Tiny},
{{c3, 1., "\!\(\*SubscriptBox[\(c\), \(3\)]\)"}, -10, 10, 1,
Appearance -> "Labeled", ImageSize -> Tiny},
{{l1, 1., "\!\(\*SubscriptBox[\(l\), \(1\)]\)"}, -10, 10, 1,
Appearance -> "Labeled", ImageSize -> Tiny},
{{l2, 1., "\!\(\*SubscriptBox[\(l\), \(2\)]\)"}, -10, 10, 1,
Appearance -> "Labeled", ImageSize -> Tiny},
{{l3, 1., "\!\(\*SubscriptBox[\(l\), \(3\)]\)"}, -10, 10, 1,
Appearance -> "Labeled", ImageSize -> Tiny}
]
Attachments: