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This small animation shows the feasibility region and the the intersection line between $a_1{x}_1+a_2{x}_2+a_3{x}_3=1$ and $b_1{x}_1+b_2{x}_2+b_3{x}_3=1$ and with $x_i\ge 0$. The light Green region is the one in which $a_1{x}_1+a_2{x}_2+a_3{x}_3\le 1$ and $b_1{x}_1+b_2{x}_2+b_3{x}_3\le 1$. The optimal solution will be on a vertix on the line of interesection between the two planes shown. Done for class HW.

This small animation shows the feasibility region which is the intersection line between unit simplex given by $x_1+x_2+x_3=1$ and plane $2x_1+3x_2=1$.

Source code used to generate the first movie is

Manipulate[ 
 len = 1.5; 
 h = a1 x1 + a2 x2 + a3 x3; 
 g = b1 x1 + b2 x2 + b3 x3; 
 
 g1 = ContourPlot3D[{h == 1, g == 1}, {x1, 0, len}, {x2, 0, len}, {x3, 0, len}, 
   PlotRange -> {{0, len}, {0, len}, {0, len}}, 
   SphericalRegion -> True, MeshStyle -> {{Thick, Blue}}, Mesh -> {{0}}, 
   Lighting -> {{"Ambient", White}} 
   ]; 
 g2 = RegionPlot3D[h <= 1 && g <= 1, {x1, 0, len}, {x2, 0, len}, {x3, 0, len}, 
   PlotRange -> {{0, len}, {0, len}, {0, len}}, 
   SphericalRegion -> True, Mesh -> 0, 
   PlotStyle -> Directive[Green, Opacity[0.2]], 
   Lighting -> {{"Ambient", White}}]; 
 
 g3 = Graphics3D[{ 
    Arrow[{{0, 0, -len}, {0, 0, len}}], 
    Text[Style[z, Bold], {0, 0, 1.1 len}], 
    Arrow[{{0, -len, 0}, {0, len, 0}}], 
    Text[Style[y, Bold], {0, 1.1 len, 0}], 
    Arrow[{{-len, 0, 0}, {len, 0, 0}}], 
    Text[Style[x, Bold], {1.1 len, 0, 0}] 
    }]; 
 Grid[{ 
   {h}, 
   {g}, 
   { 
    If[showRegion, 
     Show[g1, g2, g3, ImageSize -> 400, ImagePadding -> 5] 
     , 
     Show[g1, g3, ImageSize -> 400, ImagePadding -> 5] 
     ] 
    }}, Frame -> All], 
 {{a1, 1, "a1"}, 0, 5, .01, Appearance -> "Labeled", ImageSize -> Tiny}, 
 {{a2, 1, "a2"}, 0, 5, .01, Appearance -> "Labeled", ImageSize -> Tiny}, 
 {{a3, 1, "a3"}, 0, 5, .01, Appearance -> "Labeled", ImageSize -> Tiny}, 
 {{b1, 2, "b1"}, 0, 5, .01, Appearance -> "Labeled", ImageSize -> Tiny}, 
 {{b2, 3, "b2"}, 0, 5, .01, Appearance -> "Labeled", ImageSize -> Tiny}, 
 {{b3, 0, "b3"}, 0, 5, .01, Appearance -> "Labeled", ImageSize -> Tiny}, 
 Row[{"show feasbility region ", Checkbox[Dynamic[showRegion]]}], 
 ControlPlacement -> Left 
 ]
 

Source code used to generate the second movie is

eq1 = 2 x1 + 3 x2 == 1; 
eq2 = x1 + x2 + x3 == 1; 
len = 2; 
g2 = Graphics3D[Simplex[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}]]; 
g1 = ContourPlot3D[Evaluate@eq1, {x1, -1, 1}, {x2, -1, 1}, {x3, -1, 1}, 
   ContourStyle -> Directive[FaceForm[Yellow], Opacity[.5]], Mesh -> None, 
   Lighting -> {{"Ambient", White}}, Boxed -> True, Axes -> False]; 
g3 = Graphics3D[{ 
    Arrow[{{0, 0, -len}, {0, 0, len}}], Text[Style[z, Bold], {0, 0, 1.1 len}], 
    Arrow[{{0, -len, 0}, {0, len, 0}}], Text[Style[y, Bold], {0, 1.1 len, 0}], 
    Arrow[{{-len, 0, 0}, {len, 0, 0}}], Text[Style[x, Bold], {1.1 len, 0, 0}], 
    {Blue, Sphere[{0, 1/3, 2/3}, .05]}, 
    {Blue, Sphere[{1/2, 0, 1/2}, .05]}, 
    {Red, Thick, Line[{{0, 1/3, 2/3}, {1/2, 0, 1/2}}]} 
    }]; 
Grid[{{Column[{eq1, eq2}]}, 
  {Show[g1, g2, g3, PlotRange -> All, ImageSize -> 400, 
    SphericalRegion -> True]}}]
 

The following is the Mathematica notebook note.nb