From Fri Feb 9

Solve[{
a+2b+c==0,
3a+4b+c==0
}]

Solve[{
a+2b+c==0,
3a+4b+c==0
},{a,b,c}]

Solve[{
a+2b+c==0,
3a+4b+c==0
},{a,b}]

RowReduce[{ {1,2,1,0},
{3,4,1,0}}]

MatrixForm[{{1,0,-1,0},{0,1,1,0}}]

RowReduce[{ {1,2,1,0},
{3,4,1,0},
{3,4,1,0}}]

{{1,0,-1,0},{0,1,1,0},{0,0,0,0}}

Det[{ {1,2,1},
{3,4,1},
{0,0,0}}]

Det[{ {1,2,1,0},
{3,4,1,0},
{0,0,0,0},
{0,0,0,0}}]

MatrixForm[ RowReduce[
{{1,2,3,1,0},
{4,5,6,1,0},
{7,8,9,1,0}}]   ]

(a x + b y + c z + d == 0) /. {a-> (c+d), b-> -2c-d}

(d+(c+d) x+(-2 c-d) y+c z==0)/.{d-> 3, c-> -4}

Solve[3-x+5 y-4 z==0,z]

Plot3D[(1/4) (3-x+5 y),{x,-10,10},{y,-10,10},AxesLabel->{"x","y","z"}]

(d+(c+d) x+(-2 c-d) y+c z==0)/.{d-> -13, c-> 25}

Solve[-13+12 x-37 y+25 z==0,z]

Plot3D[(1/25) (13-12 x+37 y),{x,-10,10},{y,-10,10},AxesLabel->{"x","y","z"}]

Show[{
Plot3D[(1/4) (3-x+5 y),{x,-10,10},{y,-10,10},AxesLabel->{"x","y","z"}],
Plot3D[(1/25) (13-12 x+37 y),{x,-10,10},{y,-10,10}]
}]

{{1,2,3,1},
{4,5,6,1},
{7,8,9,1}}.{a,b,c,d}

Solve[{a+2 b+3 c+d==0,4 a+5 b+6 c+d==0,7 a+8 b+9 c+d==0},{a,b}]

visualizing the points

Clear[v1,v2,v3]
v1={1,2,3};
v2={4,5,6};
v3={7,8,9};
ListPointPlot3D[{v1,v2,v3},PlotStyle->PointSize[Large],AxesLabel->{"x","y","z"}]

Clear[ptGraphics]
ptGraphics = {Point[v1],Point[v2],Point[v3]};
Show[Graphics3D[{PointSize[0.03],ptGraphics}],Axes->True,AxesLabel->{"x","y","z"}]

Clear[L1,L2,L3]
L1=Line[{v1,v2}];
L2=Line[{v2,v3}];
L3=Line[{v1,v3}];
Show[{Graphics3D[{PointSize[0.03],ptGraphics}],
Graphics3D[{Thick,Dashing[.03],Red,L1}],Graphics3D[{Dotted,Thick,Dashing[.03],Blue,L2}], Graphics3D[{Thick,Dashing[.01],Green,L3}] },
Axes->True,AxesLabel->{"x","y","z"}]

a new set

Clear[v1,v2,v3]
v1={1,2,3};
v2={4,-1,6};
v3={4,8,9};
ListPointPlot3D[{v1,v2,v3},PlotStyle->PointSize[Large],AxesLabel->{"x","y","z"}]

Clear[ptGraphics]
ptGraphics = {Point[v1],Point[v2],Point[v3]};
Show[Graphics3D[{PointSize[0.03],ptGraphics}],
Axes->True,AxesLabel->{"x","y","z"} ]

Clear[L1,L2,L3]
L1=Line[{v1,v2}];
L2=Line[{v2,v3}];
L3=Line[{v1,v3}];
Show[{Graphics3D[{PointSize[0.03],ptGraphics}],Graphics3D[{Thick,Dashing[.03],Red,L1}],Graphics3D[{Dotted,Thick,Dashing[.03],Blue,L2}], Graphics3D[{Thick,Dashing[.01],Green,L3}] },
Axes->True,AxesLabel->{"x","y","z"}]

Clear[v1e,v2e,v3e,a,b,c,d,mat,lhs,rul]
v1e=Join[v1,{1}];
v2e=Join[v2,{1}];
v3e=Join[v3,{1}];
mat={v1e,v2e,v3e}
lhs=mat.{a,b,c,d}
rul=Solve[lhs==0][[1]]

RowReduce[mat]

Clear[x,y,z,sol,fcn]
(a x + b y + c z + d == 0) /. rul
sol =  Solve[%,z][[1]]
fcn =sol[[1,2]]

Clear[f];
f[x_,y_]:=fcn
Plot3D[f[x,y],{x,-1,5},{y,-2,10},AxesLabel->{"x","y","z"}]

Show[{    ListPointPlot3D[{v1,v2,v3},PlotStyle->PointSize[Large],AxesLabel->{"x","y","z"}], 
Plot3D[f[x,y],{x,-2,5},{y,-1,10}]    },
PlotRange->{{-2,6},{-3,11},{-1,11}}]

Show[{
Graphics3D[{PointSize[0.03],ptGraphics}],
Graphics3D[{Thick,Dashing[.03],Red,L1}],Graphics3D[{Dotted,Thick,Dashing[.03],Blue,L2}], Graphics3D[{Thick,Dashing[.01],Green,L3}] , Plot3D[f[x,y],{x,-1,5},{y,-1,10}] 
 },Axes->True,AxesLabel->{"x","y","z"}]

a 3rd set

Clear[v1,v2,v3]
v1={1,-1,3};
v2={4,5,6};
v3={2,8,9};
ListPointPlot3D[{v1,v2,v3},PlotStyle->PointSize[Large],AxesLabel->{"x","y","z"}]

Clear[ptGraphics]
ptGraphics = {Point[v1],Point[v2],Point[v3]};
Clear[L1,L2,L3]
L1=Line[{v1,v2}];
L2=Line[{v2,v3}];
L3=Line[{v1,v3}];
Show[{Graphics3D[{PointSize[0.03],ptGraphics}],Graphics3D[{Thick,Dashing[.03],Red,L1}],Graphics3D[{Dotted,Thick,Dashing[.03],Blue,L2}], Graphics3D[{Thick,Dashing[.01],Green,L3}] },
Axes->True,AxesLabel->{"x","y","z"}]

Clear[v1e,v2e,v3e,a,b,c,d,mat,lhs,rul]
v1e=Join[v1,{1}];
v2e=Join[v2,{1}];
v3e=Join[v3,{1}];
mat={v1e,v2e,v3e}
lhs=mat.{a,b,c,d}
rul=Solve[lhs==0][[1]]

RowReduce[mat]

Clear[x,y,z,sol,fcn]
(a x + b y + c z + d == 0) /. rul
sol =  Solve[%,z][[1]]
fcn =sol[[1,2]]

Clear[f];
f[x_,y_]:=fcn
Show[{
Graphics3D[{PointSize[0.03],ptGraphics}],
Graphics3D[{Thick,Dashing[.03],Red,L1}],Graphics3D[{Dotted,Thick,Dashing[.03],Blue,L2}], Graphics3D[{Thick,Dashing[.01],Green,L3}] , 
Plot3D[f[x,y],{x,-1,5},{y,-2,10}] 
 },Axes->True,AxesLabel->{"x","y","z"}]