documentclass{article}
usepackage{amsmath}
begin{subequations}
begin{alignat}{2}
&begin{multlined}[b]
{e}^{n+1} - {e}^n = Delta t {dot{e}}^{n}
+ beta Delta t^2 {ddot{e}}^{n+1} \
+ Delta t^2 left( dfrac{1}{2}-beta right) {ddot{e}}^{n}
end{multlined}\
&dfrac{1}{Delta t} left( {dot{e}}^{n+1}  -  {dot{e}}^{n} right)=dfrac{1}{2} left( {ddot{e}}^{n+1}  +  {ddot{e}}^{n}  right)
end{alignat}
end{subequations}

end{document}

documentclass{article}
usepackage{amsmath}
usepackage{array}
begin{document}
begin{align}
  omitrlap{${e}^{n+1} - {e}^n = Delta t {dot{e}}^{n} + beta Delta t^2 {ddot{e}}^{n+1}$}nonumber\
   &{}+Delta t^2 left( dfrac{1}{2}-beta right) {ddot{e}}^{n} nonumber\
  dfrac{1}{Delta t} left( {dot{e}}^{n+1}right. & {}-  left.{dot{e}}^{n} right)=dfrac{1}{2} left( {ddot{e}}^{n+1}  +  {ddot{e}}^{n}  right)
end{align}
end{document}