Better form of displaying system of equations in an enumerated list

You can use flalign* from mathtools, or aligned in equations. From your code :

documentclass[12pt]{exam}
usepackage[utf8]{inputenc}
usepackage{siunitx}
usepackage{tcolorbox}
usepackage{amsmath,amssymb,xcolor}
usepackage{mathtools}
usepackage{multicol}
usepackage{graphicx}
usepackage{enumerate}

pagestyle{plain}

usepackage{geometry}
geometry{%
  letterpaper,
  lmargin=10 mm,
  rmargin=10 mm,
  tmargin=1 cm,
  bmargin=1 cm,
  footskip=9 pt,
  headheight=12 pt}  
usepackage{titlesec}
titlelabel{thetitle.quad}

linespread{1.2}

renewenvironment{solution}{{bf Solution:}}{hfill}
firstpagefooter
  {}
  {}
  {textbf{[Turn over}}
runningfooter
  {}
  {Page thepage of numpages}
  {ifoddvalue{page}
    ifnumvalue{page}<numpagestextbf{[Turn over}fi
  fi}


begin{document}

begin{enumerate}  


item Find all existing solutions to the following linear systems, using the Gaussian elimination method with backward substitution.
begin{multicols}{2}
begin{enumerate}[(a)]
item begin{flalign*}
        4x-3y & = 10&
        8x-y & = 10&
    end{flalign*}
item begin{equation*}
left{begin{aligned}
       -3x+5y &=-22&
       3x+4y &=4& 
       4x-8y&=32&
    end{aligned}right.
    end{equation*}
item $y+z=63x-y+z=-7x+y-3z=-13$
item $x+2y=0x+y=63x-2y=8$
end{enumerate}
end{multicols}
end{enumerate}
end{document}