a=ContourPlot[(x-(2/3))(y+(1/4)),{x,-3,3},{y,-3,3},ContourShading->False,ContourLabels->True,GridLines->Automatic,Contours->Table[-8+k,{k,1,16}]]

b=RegionPlot[And[-1≤x≤2,y≤x+1, -(1/3)(x+1)≤y≤5-3x],{x,-3,3},{y,-3,3},PlotPoints->50,PlotStyle->LightBlue]

bgrid=RegionPlot[And[-1≤x≤2,y≤x+1, -(1/3)(x+1)≤y≤5-3x],{x,-3,3},{y,-3,3},PlotPoints->50,PlotStyle->LightBlue,GridLines->Automatic]

Show[bgrid,Plot[{-(1/3)(x+1),x+1,5-3x},{x,-3,3},PlotStyle->{{Thick,Blue},{Thick,Blue},{Thick,Blue}}],Graphics[{PointSize[Large],Black,Point[{{-1,0},{1,2},{2,-1}}]}]]

Show[a,b]

D[(x-(2/3))(y[x]+(1/4)),x]

Expand[%/.y'[x]->-3]

%/.{y[x]->5-3x}

Solve[%==0,x]

5-3%[[1,1,2]]

k=(x-(2/3))(y+(1/4))/.{x->29/24,y->11/8}

N[{{29/24,11/8},k}]

ContourPlot[(x-(2/3))(y+(1/4))==k,{x,0,3},{y,0,3},ContourStyle->{Thick,Red}];

Show[a,b,%,Graphics[{PointSize[Large],Black,Point[{29/24,11/8}]}]]