Remove vertical space before and after align with enumerate / make multicols align to the top

Question

MWE:

\documentclass{article}
\usepackage[top=3cm,bottom=3cm,left=3.2cm,right=3.2cm,headsep=10pt,a4paper]{geometry} 
\usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage{enumitem} 
\usepackage{multicol}
\setlength{\columnsep}{1cm} 
\setlength{\columnseprule}{1pt}
\allowdisplaybreaks


\begin{document}


\begin{multicols*}{2}

\begin{enumerate}[label=(\alph*)]
\item \begin{align*}
2x+6&=12x-8\\
2x-12x&=-8-6\\
-10x&=-14\\
x&=\dfrac{-14}{-10}\\
&=1\dfrac{2}{5}
\end{align*}
\item \begin{align*}
4x-5&=6(7)\\
&=42\\
4x&=42+5\\
&=47\\
x&=\dfrac{47}{4}\\
&=11\dfrac{3}{4}
\end{align*}
\item \begin{align*}
5(6x-2)&=3(9x+1)\\
30x-10&=27x+3\\
30x-27x&=3+10\\
3x&=13\\
x&=\dfrac{13}{3}\\
&=4\dfrac{1}{3}
\end{align*}
\item \begin{align*}
\dfrac{3(-2x+5)-4(8x+1)}{12}&=7\\
\dfrac{-6x+15-32x-4}{12}&=7\\
\dfrac{-38x+11}{12}&=7
\end{align*}\\
\begin{align*}
-38x+11&=12(7)\\
&=84\\
-38x&=84-11\\
&=73\\
x&=\dfrac{73}{-38}\\
&=-1\dfrac{35}{38}
\end{align*}
\item \begin{align*}
3&=7(x+5)\\
3&=7x+35\\
-7x&=35-3\\
&=32\\
x&=\dfrac{32}{-7}\\
&=-4\dfrac{4}{7}
\end{align*}
\item \begin{align*}
4(5x+6)&=6(3x-2)\\
20x+24&=18x-12\\
20x-18x&=-12-24\\
2x&=-36\\
x&=\dfrac{-36}{2}\\
&=-18
\end{align*}
\end{enumerate}


\end{multicols*}

\end{document}

I am trying to create a solutions page, and would like to remove the vertical space such that (a) is on the same line as the first equation, and there is no vertical space after the last equation, before (b).

Also, the second column shows that LaTex is filling up the entire column with my equations but I would like it to be aligned to the top.

How might I achieve both of these?

Thank you.

Answer 1

In this code I define a solutions environment, based on enumerate, but with the \item command patched with the help of the etoolbox package. I also changed some of the computation results, expressed in theinteger part fractionary part style with smaller fractions, either in the form, say, \tfrac{3}{4} or in the form \sfrac{3}{4} from the xfrac package:

\documentclass{article}
\usepackage[top=3cm,bottom=5cm,left=3.2cm,right=3.2cm,headsep=10pt,a4paper]{geometry}
\usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage{enumitem}
\usepackage{multicol}
\setlength{\columnsep}{1cm}
\setlength{\columnseprule}{1pt}
\allowdisplaybreaks
\usepackage{xfrac}
\usepackage{etoolbox}
\newlist{solutions}{enumerate}{1}
\setlist[solutions]{label =(\alph*)}
\AtBeginEnvironment{solutions}{%\pretocmd{\item}{\renewcommand{\abovedisplayskip}{0pt}}{}{}
\apptocmd{\item}{\mbox{}\vspace*{\dimexpr-\baselineskip-\abovedisplayskip\relax}}{}{}}
%{\apptocmd{\item}{\leavevmode\vspace*{\dimexpr-\baselineskip-\abovedisplayskip\relax}}{}{}}
\raggedbottom
\begin{document}

\begin{multicols*}{2}

  \begin{solutions}
    \item \begin{align*}
    2x+6&=12x-8\\
    2x-12x&=-8-6\\
    -10x&=-14\\
    x&=\dfrac{-14}{-10}\\
    &=1\mkern1mu\tfrac{2}{5}
    \end{align*}
    %
    \item \begin{align*}
    4x-5&=6(7)\\
    &=42\\
    4x&=42+5\\
    &=47\\
    x&=\dfrac{47}{4}\\
    &=11\mkern1mu\tfrac{3}{4}
    \end{align*}
    %
    \item \begin{align*}
    5(6x-2)&=3(9x+1)\\
    30x-10&=27x+3\\
    30x-27x&=3+10\\
    3x&=13\\
    x&=\dfrac{13}{3}\\
    &=4\mkern1mu\tfrac{1}{3}
    \end{align*}
    %
    \item
    \begin{align*}
      \dfrac{3(-2x+5)-4(8x+1)}{12} & =7 \\
      \dfrac{-6x+15-32x-4}{12} & =7 \\
      \dfrac{-38x+11}{12} & =7 \\
    \end{align*}\vskip-9ex
    \begin{align*}
      -38x+11 & =12(7) =84 \\
      -38x & =84-11 \\
              & =73 \\
      x & =\dfrac{73}{-38} \\
              & =-1\sfrac{35}{38}
    \end{align*}
    %
    \item \begin{align*}
    3&=7(x+5)\\
    3&=7x+35\\
    -7x&=35-3\\
    &=32\\
    x&=\dfrac{32}{-7}\\
    &=-4\sfrac{4}{7}
    \end{align*}
    %
    \item \begin{align*}
    4(5x+6)&=6(3x-2)\\
    20x+24&=18x-12\\
    20x-18x&=-12-24\\
    2x&=-36\\
    x&=\dfrac{-36}{2}\\
    &=-18
    \end{align*}
  \end{solutions}
\end{multicols*}

\end{document}