How can we use tikz package to draw a rectangle and fill it with a color for the equations?

You want to use tcolorbox:

documentclass{article}
usepackage{amssymb}
usepackage{tcolorbox}

begin{document}

    begin{tcolorbox}[colback=blue!10,text width=5cm]
begin{equation}
centering  f(x)=ax+b
end{equation}
    end{tcolorbox}
end{document}

with tcbset{colframe=white}:

in tikz(but tcolorobx i think is the better one):

documentclass{article}
usepackage{amssymb}
usepackage{tikz}

begin{document}

begin{tikzpicture}

node [fill=cyan!20,text width=4cm] at (0,0) {begin{equation}f(x)=ax+bend{equation}};

    end{tikzpicture}
end{document}

documentclass{article}
usepackage{amsmath}
usepackage[customcolors]{hf-tikz}

begin{document}
hfsetfillcolor{blue!10}
hfsetbordercolor{blue}
begin{equation}label{e:barwq2}
tikzmarkin{b}  f(x)=ax+btikzmarkend{b}
end{equation}
end{document}

for white borders

documentclass{article}
usepackage{amsmath}
usepackage[customcolors]{hf-tikz}

begin{document}
hfsetfillcolor{blue!20}
hfsetbordercolor{white}
begin{equation}label{e:barwq2}
tikzmarkin{b}  f(x)=ax+btikzmarkend{b}
end{equation}
end{document}