Shading to indicate the solution set to a system of two linear inequalities in x and y

Question

I am trying to show graphically the solution to the system of linear inequalities

\begin{equation*}

\begin{cases}
7x + 3y \geq 21 \\
-x + 3y < -3
\end{cases}
.
\end{equation*}

There are some modifications that I want to the display. First, I use pattern=north east lines for the solution set to 7x + 3y \geq 21. The lines are too close. I do not want the boundary. Second, I get an error when I tried to compile the code with pattern=south east lines. Third, I do not want these lines indicating the solution sets to interfere with the tick marks.

\documentclass{amsart}

\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings,backgrounds,patterns}

\usepackage{pgfplots}
\pgfplotsset{compat=1.11}


\begin{document}

\begin{center}\Large{\bf Algebra 2}\end{center}\vskip0.3in

\noindent \hspace*{\fill}
\begin{tikzpicture}
\begin{axis}[width=6in,grid=both,grid style={line width=.1pt, draw=gray!10},
    axis equal image,
    axis lines=middle,
    xmin=-11,xmax=11,
    ymin=-11,ymax=11,
    restrict y to domain=-11:11,
    xtick={},ytick={},
    ticklabel style={font=\tiny,fill=white},
    enlargelimits={abs=0.25cm},
    axis line style={latex-latex},
    axis line style={shorten >=-7.5pt, shorten <=-7.5pt},
    xlabel=$x$,
    ylabel=$y$,
    xlabel style={at={(ticklabel* cs:1)},anchor=north west},
    ylabel style={at={(ticklabel* cs:1)},anchor=south west}
]


\coordinate (P) at (-9/7,10);
\coordinate (Q) at (10,10);
\coordinate (R) at (10,-10);
\coordinate (S) at (51/7,-10);

\draw[pattern=north east lines] (P) -- (Q) -- (R) -- (S);


\coordinate (P1) at (-10,-13/3);
\coordinate (Q1) at (10,7/3);
\coordinate (R1) at (10,-10);
\coordinate (S1) at (-10,-10);

%\draw[pattern=south east lines] (P1) -- (Q1) -- (R1) -- (S1);

\addplot[latex-latex,samples=2,domain=-1.5:7.5,blue] {(-7/3)*x + 7};
\addplot[latex-latex,samples=2,domain=-10:10,dashed,green] {(1/3)*x -1};

\node[anchor=north west] at (axis description cs:1,0.5) {$x$};
\node[anchor=north west] at (axis description cs:0.5,1) {$y$};

\end{axis}
\end{tikzpicture}

\end{document}

Answer 1

Don't really like this, but I think this is what you asked for:

Notes:

• Patterns are not easily changeable. I have used the code form Can I control the 'density' of a pattern in TikZ? to allow one to change the spacing between the lines. The density of the lines below can be changed via line space=<dimen>.

• The border along the pattern was disabled via draw=none option.

• Adding axis on top option allows the axis and tick labels to be on top.

• There is no south east lines pattern, but there is a north west lines . I have applied the my north west lines as it was desired to alter the spacing between the lines.

Code:

\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings,backgrounds,patterns}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
%% https://tex.stackexchange.com/questions/29808/can-i-control-the-density-of-a-pattern-in-tikz
\pgfdeclarepatternformonly[\LineSpace]{my north east lines}{\pgfqpoint{-1pt}{-1pt}}{\pgfqpoint{\LineSpace}{\LineSpace}}{\pgfqpoint{\LineSpace}{\LineSpace}}%
{
    \pgfsetlinewidth{0.4pt}
    \pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
    \pgfpathlineto{\pgfqpoint{\LineSpace + 0.1pt}{\LineSpace + 0.1pt}}
    \pgfusepath{stroke}
}
\pgfdeclarepatternformonly[\LineSpace]{my north west lines}{\pgfqpoint{-1pt}{-1pt}}{\pgfqpoint{\LineSpace}{\LineSpace}}{\pgfqpoint{\LineSpace}{\LineSpace}}%
{
    \pgfsetlinewidth{0.4pt}
    \pgfpathmoveto{\pgfqpoint{0pt}{\LineSpace}}
    \pgfpathlineto{\pgfqpoint{\LineSpace + 0.1pt}{-0.1pt}}
    \pgfusepath{stroke}
}
\newdimen\LineSpace
\tikzset{
    line space/.code={\LineSpace=#1},
    line space=3pt
}
\begin{document}
\begin{center}\Large{\bf Algebra 2}\end{center}\vskip0.3in
\noindent \hspace{\fill}
\begin{tikzpicture}
\begin{axis}[width=6in,grid=both,grid style={line width=.1pt, draw=gray!10},
    axis equal image,
    axis lines=middle,
    xmin=-11,xmax=11,
    ymin=-11,ymax=11,
    restrict y to domain=-11:11,
    xtick={},ytick={},
    ticklabel style={font=\tiny,fill=white},
    enlargelimits={abs=0.25cm},
    axis line style={latex-latex},
    axis line style={shorten >=-7.5pt, shorten <=-7.5pt},
    xlabel=$x$,
    ylabel=$y$,
    xlabel style={at={(ticklabel cs:1)},anchor=north west},
    ylabel style={at={(ticklabel* cs:1)},anchor=south west},
    axis on top,
]
\coordinate (P) at (-9/7,10);
\coordinate (Q) at (10,10);
\coordinate (R) at (10,-10);
\coordinate (S) at (51/7,-10);
%% Adjust density of the lines by changing the line space= value.
\draw[pattern=my north west lines, draw=none, line space=10pt] (P) -- (Q) -- (R) -- (S);
\coordinate (P1) at (-10,-13/3);
\coordinate (Q1) at (10,7/3);
\coordinate (R1) at (10,-10);
\coordinate (S1) at (-10,-10);
%\draw[pattern=south east lines] (P1) -- (Q1) -- (R1) -- (S1);
\addplot[latex-latex,samples=2,domain=-1.5:7.5,blue, ultra thick] {(-7/3)x + 7};
\addplot[latex-latex,samples=2,domain=-10:10.4,dashed,orange, thick] {(1/3)x -1};
\node[anchor=north west] at (axis description cs:1,0.5) {$x$};
\node[anchor=north west] at (axis description cs:0.5,1) {$y$};
\end{axis}
\end{tikzpicture}
\end{document}