You did not provide 't20' or a description of its structure, so I had to make my own.
Hopefully it is compatible with yours:
t20 = Table[
"inert"[{x, y, Sin[x Sin[y]]}], {x, 1, 10, .1}, {y, 1, 10, .1}];
t20 = Cases[t20, "inert"[a___] :> a, Infinity];
I think I achieved what you wanted with the following:
Manipulate[
If[XorY == "y=c",
Grid[{{ListLinePlot[Cases[t20, {x_, c, z_} :> {x, z}],
AxesLabel -> {"x", "z"}, PlotStyle -> Thick,
ImageSize -> {275, 275}],
ListPlot3D[t20, Mesh -> {{0}, {{c, Thick}}},
ImageSize -> {275, 275}]}}],
Grid[{{ListLinePlot[Cases[t20, {c, y_, z_} :> {y, z}],
AxesLabel -> {"y", "z"}, PlotStyle -> Thick,
ImageSize -> {275, 275}],
ListPlot3D[t20, Mesh -> {{{c, Thick}}, {0}},
ImageSize -> {275, 275}]}}]], {{XorY, "x=c",
"direction"}, {"x=c", "y=c"},
ControlType -> Setter}, {{c, 1., "c"}, 1., 10., .1,
Appearance -> "Labeled"}]
A matrix uses the Part[] or [[]] syntax. However you do not seem to have a matrix. You seem to have a list constaining lists of three coordinates.
That is, instead of having something such as {{1,2,3},{4,5,6}}, you have the equivalent information in a different form: {{1,1,1},{1,2,2},{1,3,3},{2,1,4},{2,2,5},{2,3,6}}.
Cases[] would be a way to extract just the coordinates that you are interested in, such as the ones where the first coordinate (the 'x' coordinate) is equal to a constant 'c'.
I would try to present this better but I am out of time for now. I hope it helps you.
