A CliffordBasic solution to an R4 linear system a x + b y = c, using wedge products to solve.  Also includes mmacell output to embed the solution in the book as Mathematica input and output.

Solve: a x + b y = c

<< CliffordBasic`;
$SetSignature={4,0};

Import["https://raw.githubusercontent.com/jkuczm/MathematicaCellsToTeX/master/NoInstall.m"]

(*Import["https://raw.githubusercontent.com/jkuczm/MathematicaSyntaxAnnotations/master/BootstrapInstall.m"]
Import["https://raw.githubusercontent.com/jkuczm/MathematicaCellsToTeX/master/BootstrapInstall.m"]*)

ClearAll[a, b, c, iab, aWedgeB, cWedgeB, aWedgeC, x, y]
a = e[1] + e[2];
b = e[1] + e[4];
c = e[1] + 2 e[2] - e[4]; 

aWedgeB = OuterProduct[a,b];
cWedgeB = OuterProduct[c,b];
aWedgeC = OuterProduct[a,c];

(* 1/aWedgeB *)
iab = aWedgeB / GeometricProduct[aWedgeB, aWedgeB]; 
x = GeometricProduct[ iab, cWedgeB];
y = GeometricProduct[ iab, aWedgeC];

{{"a ⋀ b = ", aWedgeB},{"c ⋀ b = ", cWedgeB}, 
{"a ⋀ c = ", aWedgeC},{"x = ", x},{"y = ", y}
} // Grid

Generate cells

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junk.  first experimentation with mmaCell

testCell=Cell[BoxData[MakeBoxes[Subscript[x,1]==(-b±Sqrt[b^2-4 a c])/(2 a)]],"Input"];
testCell//CellPrint
CellToTeX[testCell]

(Subscript[x, 1])==((-b±(Sqrt[(b^2)-4 a c]))/(2 a))

testCell=Cell[BoxData[MakeBoxes[Subscript[x,1]==(-b±Sqrt[b^2-4 a c])/(2 a)]],"Input"];
testCell//CellPrint
CellToTeX[testCell]

(Subscript[x, 1])==((-b±(Sqrt[(b^2)-4 a c]))/(2 a))

testCell=Cell[BoxData[MakeBoxes[aWedgeB 
cWedgeB;
aWedgeC ;
x;
y]],"Output"];
testCell//CellPrint
CellToTeX[testCell]

ClearAll[a,b,c,iab,aWedgeB,cWedgeB,aWedgeC,x,y];a=e[1]+e[2];b=e[1]+e[4];c=e[1]+2 e[2]-e[4];aWedgeB=OuterProduct[a,b];cWedgeB=OuterProduct[c,b];aWedgeC=OuterProduct[a,c];iab=(aWedgeB/(GeometricProduct[aWedgeB,aWedgeB]));x=GeometricProduct[iab,cWedgeB];y=GeometricProduct[iab,aWedgeC];({{"\!\(\*StyleBox[\"a\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"⋀\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"b\",FontWeight->\"Plain\"]\) = ", aWedgeB}, {"\!\(\*StyleBox[\"c\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"⋀\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"b\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)= ", cWedgeB}, {"\!\(\*StyleBox[\"a\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"⋀\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"c\",FontWeight->\"Plain\"]\) = ", aWedgeC}, {"x = ", x}, {"y = ", y}})

ClearAll[a,b,c,iab,aWedgeB,cWedgeB,aWedgeC,x,y] a=e[1]+e[2];b=e[1]+e[4];c=e[1]+2 e[2]-e[4];aWedgeB=OuterProduct[a,b];cWedgeB=OuterProduct[c,b];aWedgeC=OuterProduct[a,c];iab=(aWedgeB/(GeometricProduct[aWedgeB,aWedgeB]));x=GeometricProduct[iab,cWedgeB];y=GeometricProduct[iab,aWedgeC];({{"\!\(\*StyleBox[\"a\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"⋀\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"b\",FontWeight->\"Plain\"]\) = ", aWedgeB}, {"\!\(\*StyleBox[\"c\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"⋀\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"b\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)= ", cWedgeB}, {"\!\(\*StyleBox[\"a\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"⋀\",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\" \",FontWeight->\"Plain\"]\)\!\(\*StyleBox[\"c\",FontWeight->\"Plain\"]\) = ", aWedgeC}, {"x = ", x}, {"y = ", y}})

"\\begin{mmaCell}[moredefined={a, b, c, iab, aWedgeB, cWedgeB, aWedgeC, x, y, e, OuterProduct, GeometricProduct}]{Input}\n  ClearAll[a,b,c,iab,aWedgeB,cWedgeB,aWedgeC,x,y];
a=e[1]+e[2];
b=e[1]+e[4];
c=e[1]+2e[2]-e[4];
aWedgeB=OuterProduct[a,b];
cWedgeB=OuterProduct[c,b];
aWedgeC=OuterProduct[a,c];
iab=\\mmaFrac{aWedgeB}{GeometricProduct[aWedgeB,aWedgeB]};
x=GeometricProduct[iab,cWedgeB];
y=GeometricProduct[iab,aWedgeC];
a \\(\\pmb{\\wedge}\\) b = \taWedgeB\n  c \\(\\pmb{\\wedge}\\) b = \tcWedgeB\n  a \\(\\pmb{\\wedge}\\) c = \taWedgeC\n  \"x = \"\tx\n  \"y = \"\ty\n\\end{mmaCell}"