(1) Basic
\begin{align}
y = f(x) \\ %automatically numbering, "\\" means "change the line"
y = g(x) \nonumber \\ % "\nonumber" means "do not numbering this equation"
y = \frac{1}{x} \nonumber
\end{align}(2) Alignment
\begin{align}
y &= f(x) \nonumber \\
&= T_{0}+ (T_{0}-T_{h})e^{-x} % "&"s are automatically aligned them among themselves.
\end{align}(e.g.1) Simple example: y=1-x
\documentclass[tikz,border=10pt]{standalone} %border: 여백 추가 (현재 상하좌우 10pt)
\usepackage{tikz}
\usepackage{amsmath} % 수식 표현을 위한 패키지
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[scale=4]
% Draw axes
\draw[->] (-0.2, 0) -- (2.1, 0) node[right] {$x_1$};
\draw[->] (0, -0.2) -- (0, 2.1) node[above] {$x_2$};
\draw[thick] (1, 0.02) -- (1, -0.02) node[below left] {$1$}; % x1-axis intersection
\draw[thick] (-0.02, 1) -- (0.02, 1) node[below left] {$1$}; % x2-axis intersection
\draw[dashed] (1, -0.2) -- (1, 1.9) node[right] {};
\draw[dashed] (-0.2, 1) -- (1.9, 1) node[right] {};
\draw[thick, gray , domain=0:2, samples=100]
plot(\x, {2 -\x})
node[yshift=1.5cm]{$x_1+x_2<2$};
\fill[pattern=north east lines, pattern color=gray, opacity=0.7]
(0, 1) -- (1, 0) -- (1, 1) -- cycle;
\fill[blue] (1, 1) circle (0.5pt)
\draw[thick, blue, domain=-0.2:1.3, samples=100]
plot (\x, {1 - \x}) ;
\node[below left] at (0, 0) {$(0, 0)$};
\end{tikzpicture}
\end{document}
(e.g.2) More complicated example
\documentclass[tikz,border=10pt]{standalone} %border: 여백 추가 (현재 상하좌우 10pt)
\usepackage{tikz}
\usepackage{amsmath} % 수식 표현을 위한 패키지
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[scale=4]
% Draw axes
\draw[->] (-0.2, 0) -- (2.1, 0) node[right] {$x_1$};
\draw[->] (0, -0.2) -- (0, 2.1) node[above] {$x_2$};
\draw[thick] (1, 0.02) -- (1, -0.02) node[below left] {$1$}; % x1-axis intersection
\draw[thick] (-0.02, 1) -- (0.02, 1) node[below left] {$1$}; % x2-axis intersection
\draw[dashed] (1, -0.2) -- (1, 1.9) node[right] {};
\draw[dashed] (-0.2, 1) -- (1.9, 1) node[right] {};
\draw[thick, gray , domain=0:2, samples=100]
plot(\x, {2 -\x})
node[yshift=1.5cm]{$x_1+x_2<2$};
\fill[pattern=north east lines, pattern color=gray, opacity=0.7]
(0, 1) -- (1, 0) -- (1, 1) -- cycle;
\fill[blue] (1, 1) circle (0.5pt)
node[above right,scale=0.8]{soft};
\draw[thick, blue, domain=-0.2:1.3, samples=100]
plot (\x, {1 - (0^2) - \x}) ;
\draw[thick, red, domain=-0.2:1, samples=100]
plot(\x, {1 - (0^2) / (1-\x)})
node[above, xshift=-2cm, yshift=0cm,scale=0.8] {collinear};
\draw[thick, red, domain=-0.2:1, samples=100]
plot(1, \x)
node[right, xshift=0cm, yshift=-2cm,scale=0.8] {collinear};
\node[below left] at (0, 0) {$(0, 0)$};
\end{tikzpicture}
\end{document}
(e.g.3) More more complicated example
\documentclass[tikz,border=10pt]{standalone} %border: 여백 추가 (현재 상하좌우 10pt)
\usepackage{tikz}
\usepackage{amsmath} % 수식 표현을 위한 패키지
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[scale=4]
% Draw axes
\draw[->] (-0.2, 0) -- (2.1, 0) node[right] {$x_1$};
\draw[->] (0, -0.2) -- (0, 2.1) node[above] {$x_2$};
\draw[thick] (0.75, 0.02) -- (0.75, -0.02) node[below] {$1 - \frac{\mu^2}{s}$}; % x1-axis intersection
\draw[thick] (-0.02, 0.75) -- (0.02, 0.75) node[left] {$1 - \frac{\mu^2}{s}$}; % x2-axis intersection
\draw[dashed] (1, -0.2) -- (1, 1.9) node[right] {};
\draw[dashed] (-0.2, 1) -- (1.9, 1) node[right] {};
\draw[thick, gray , domain=0:2, samples=100]
plot(\x, {2 -\x})
node[yshift=1.5cm]{$x_1+x_2<2$};
\fill[pattern=north east lines, pattern color=gray, opacity=0.7]
plot[domain=0:0.75, samples=100] (\x, {1 - (0.5^2) - \x}) --
plot[domain=0.75:0, samples=100] (\x, {1 - (0.5^2) / (1-\x)}) -- cycle;
% First equation: x_1 + x_2 = 1 - \mu^2 / s
\draw[thick, blue, domain=-0.2:1.3, samples=100]
plot (\x, {1 - (0.5^2) - \x})
node[below right, xshift=-0.2cm, yshift=0.8cm,scale=0.6] {$z=+1:x_1 + x_2 = 1 - \frac{\mu^2}{s}$};
% Second : (1-x_1)(1-x_2) = \mu^2 / s
\draw[thick, red, domain=-0.2:0.8, samples=100]
plot(\x, {1 - (0.5^2) / (1-\x)})
node[above right, xshift=-1.5cm, yshift=3.5cm,scale=0.6] {$z=-1:(1-x_1)(1-x_2) =\frac{\mu^2}{s}$};
\draw[thick, red, domain=1.3:1.95, samples=100]
plot(\x, {1 - (0.5^2) / (1-\x)});
% Third : z=0.5 case
\draw[dashed, domain=-0.2:1, samples=100]
plot(\x, { ( \x - (1-0.25) )/ (0.25 * \x - 1) });
\draw[dashed, domain=-0.2:1, samples=100]
plot(\x, { ( \x - (1-0.25) )/ (0.5*(1-0.1) * \x - 1) });
\draw[dashed, domain=-0.2:1, samples=100]
plot(\x, { ( \x - (1-0.25) )/ (0.5*(1+0.3) * \x - 1) });
\draw[dashed, domain=-0.2:0.9, samples=100]
plot(\x, { ( \x - (1-0.25) )/ (0.5*(1+0.7) * \x - 1) });
% Add origin
\node[below left] at (0, 0) {$(0, 0)$};
\end{tikzpicture}
\end{document}
(Optional) Upload your image file on your overleaf cloud
\begin{figure}[h!]
\centering
\includegraphics[width=0.5\textwidth]{Figures/figure1.pdf}
\caption{Caption}
\end{figure}
\FloatBarrier %(Optional) This line prevents the pictures from being placed in an order I didn't intend.