In[132]:=
<<Graphics`Shapes`
In[133]:=
Solve[{x^2+y^2+z^2\[Equal]1,x+y+z\[Equal]0},{x,y}]
Out[133]=
\!\({{x \[Rule] 1\/2\ \((\(-z\) - \@\(2 - 3\ z\^2\))\), 
      y \[Rule] 1\/2\ \((\(-z\) + \@\(2 - 3\ z\^2\))\)}, {x \[Rule] 
        1\/2\ \((\(-z\) + \@\(2 - 3\ z\^2\))\), 
      y \[Rule] 1\/2\ \((\(-z\) - \@\(2 - 3\ z\^2\))\)}}\)
In[134]:=
\!\(\(p1[z_] := {1\/2\ \((\(-z\) - \@\(2 - 3\ z\^2\))\), 
        1\/2\ \((\(-z\) + \@\(2 - 3\ z\^2\))\), z};\)\[IndentingNewLine]
  \(p2[z_] := {1\/2\ \((\(-z\) + \@\(2 - 3\ z\^2\))\), 
        1\/2\ \((\(-z\) - \@\(2 - 3\ z\^2\))\), z};\)\[IndentingNewLine]
  \)
In[136]:=
\!\(\(Show[{\[IndentingNewLine]WireFrame[
          Graphics3D[
            Sphere[1, 20, 
              20]]], \[IndentingNewLine]Graphics3D[{\[IndentingNewLine]\
RGBColor[1, 0, 0], \[IndentingNewLine]Thickness[
              0.009], \[IndentingNewLine]Line[
              Table[p1[z], {z, \(-\@\(2\/3\)\), \@\(2\/3\), 
                  0.01}]], \[IndentingNewLine]Line[
              Table[p2[z], {z, \(-\@\(2\/3\)\), \@\(2\/3\), 
                  0.01}]], \[IndentingNewLine]Text[
              StyleForm["\<x+y+z==0\>", Section, 
                FontColor \[Rule] RGBColor[1, 0, 0]], {1, 0.2, 1}, {\(-1\), 
                0}]}]\[IndentingNewLine]}, Boxed \[Rule] False, 
      Axes \[Rule] True, ViewPoint -> {8.043, \ \(-2.956\), \ 1.784}, 
      Boxed \[Rule] True, 
      AxesEdge \[Rule] {{1, \(-1\)}, {1, \(-1\)}, {1, 1}}, 
      AxesLabel \[Rule] {StyleForm[x, Section], StyleForm[y, Section], 
          StyleForm[z, Section]}, 
      Ticks \[Rule] {{\(-1\), \(- .5\), 0,  .5, 1}, {\(-1\), \(- .5\), 
            0,  .5, 1}, {\(-1\), \(- .5\), 0,  .5, 1}}, 
      PlotLabel \[Rule] 
        StyleForm[TraditionalForm[x^2 + y^2 + z^2 \[Equal] 1]\ , Section, 
          FontColor -> RGBColor[0, 0, 0]]]\ ;\)\)