I am a beginner in latex , but I am learning as I go, I want to plot the following linear programing problem:

(plot : to make a graphic)

f(x,y) = 2x - 3y -> max



x + y <= 12



x,y >=0

here is what I have so far, but I can`t find any good answers on how to draw in latex:

documentclass[12pt]{article}

usepackage{amsmath}

usepackage{graphicx}

usepackage{pgfplots}

usepackage{tikz}

usepackage{hyperref}

usepackage[latin1]{inputenc}





title{Linear Programming}

date{12/27/18}



begin{document}

maketitle



   bigbreak

   a) begin{cases} f(x,y) = 2x - 3y  rightarrow max \x + y   leq 12 \ x , y geq 0 end{cases}



bigbreak

bigbreak



begin{tikzpicture}

begin{axis}

  [

    xmin=-20,xmax=20, 

    ymin=-20,ymax=20,

    grid=both,

    grid style={line width=.1pt, draw=darkgray!10},

    major grid style={line width=.2pt,draw=darkgray!50},

    axis lines=middle,

    minor tick num=4,

    enlargelimits={abs=0.5},

    axis line style={latex-latex},

    samples=100,

    domain = -20:20,

  ]

  addplot[red] {x + y <= 12}; 



end{axis}

end{tikzpicture}





end{document}

Now the problem is I am only drawing lines when I should be drawing semi planes ( x + y <=12 with x and y >0 , actually a filled triangle)

What I want to achieve:

Can you help me out?

You could use filldraw from vanilla tikz to draw + fill the region. Here's an example

documentclass[12pt]{article}

usepackage{amsmath}

usepackage{graphicx}

usepackage{pgfplots}

usepackage{tikz}

usepackage{hyperref}

usepackage[latin1]{inputenc}



pgfplotsset{compat=1.12}

usetikzlibrary{patterns}





title{Linear Programming}

author{Random Numbers}

date{12/27/18}



begin{document}

maketitle



[

begin{cases} f(x,y) = 2x - 3y  rightarrow max \x + y   leq 12 \ x , y geq 0 end{cases}

]

bigbreak

bigbreak



begin{tikzpicture}

begin{axis}

  [

    xmin=-5,xmax=15,

    ymin=-5,ymax=15,

    grid=both,

    grid style={line width=.1pt, draw=darkgray!10},

    major grid style={line width=.2pt,draw=darkgray!50},

    axis lines=middle,

    minor tick num=4,

    enlargelimits={abs=0.5},

    axis line style={latex-latex},

    samples=100,

    domain = -20:20,

  ]

  %addplot[red] {x + y  12};

  filldraw[blue, pattern=north west lines, pattern color=blue] (0, 0) -- (0, 12) -- (12, 0) -- cycle;

end{axis}

end{tikzpicture}

A PSTricks solution only for fun purposes.

documentclass[pstricks,border=12pt,12pt]{standalone}

usepackage{pst-plot}

begin{document}

pspicture(-1,-1)(13,13)

psaxes[Dx=2,Dy=2]{->}(0,0)(-1,-1)(12.5,12.5)[$x$,0][$y$,90]

pspolygon[fillstyle=vlines,hatchsep=3pt,hatchangle=30](0,0)(12,0)(0,12)

pcline[nodesep=-1](12,0)(0,12)

endpspicture

end{document}

Miscellaneous

documentclass[pstricks,border=12pt]{standalone}

defRightPart

{

    psline(0.5,2.7)(!6.5 3.5 82 PtoCab)

    psbezier(1.2,1.3)(1.3,1.0)(2.0,1.0)

    psbezier(3.0,1.0)(3.0,2.2)(!6.5 3.5 73 PtoCab)

    psellipticarcn(6.5,3.5){(!6.5 3.5 73 PtoCab)}{(!6.5 3.5 73 neg PtoCab)}

    psbezier(4,-2)(4,0)(2.2,-1.8)

    psbezier(1.5,-1)(1,-1)(!6.5 3.5 -90 PtoCab)

}



begin{document}

begin{pspicture}(-7,-4)(7,4)

pscustom[dimen=m,linejoin=0,fillstyle=vlines]

{

    RightPart

    reversepath

    scale{-1 1}

    RightPart

    closepath

}

end{pspicture}

end{document}

Very similar to caverac's nice answer except that this really uses a function, i.e. also works when the constraint is more complex.

documentclass[12pt]{article}

usepackage{amsmath}

usepackage{tikz}

usetikzlibrary{patterns}

usepackage{pgfplots}

pgfplotsset{compat=1.16}

usepackage{hyperref}

usepackage[latin1]{inputenc}





title{Linear Programming}

date{12/27/18}



begin{document}

maketitle



   a) $begin{cases} f(x,y) = 2x - 3y  rightarrow text{max} \x + y   leq 12 \ x ,

   y geq 0 end{cases}$





begin{tikzpicture}

begin{axis}

  [

    xmin=-20,xmax=20, 

    ymin=-20,ymax=20,

    grid=both,

    grid style={line width=.1pt, draw=darkgray!10},

    major grid style={line width=.2pt,draw=darkgray!50},

    axis lines=middle,

    minor tick num=4,

    enlargelimits={abs=0.5},

    axis line style={latex-latex},

    samples=100,

    domain = -20:20,

  ]

  addplot[red,pattern=north east lines,pattern color=red] {12-x} -- (-20,-8) closedcycle; 



end{axis}

end{tikzpicture}

end{document}