Horizontal alignment in minipages

I'm having another trouble with horizontal alignment. Someone suggested in my last question to use minipages intead of multicol. I did use and everything was perfect, but I'm trying now to make three columns and I'm having trouble aligning the first line of the first column with the rest of the columns (take a look at the image) Here is the code I'm using:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[brazil]{babel}
\usepackage{amsmath}
\subsection{Estabilidade de cada ponto de equilíbrio em relação a $\mu$}
\begin{minipage}[t]{0.3\linewidth}
$(\overline{x_1},\overline{x_2}) = (0, 0)$
$\begin{cases} \lambda_1 = 0\\ \lambda_2 = -1\\ \end{cases}$
\begin{align*}
\lambda_1 &= 0 &\forall &\mu \in \Re
\end{align*}
\end{minipage}
\vrule{}
\begin{minipage}[t]{0.3\linewidth}
$(\overline{x_1},\overline{x_2}) = (\sqrt{\mu}, 0)$
$\begin{cases} \lambda_1 = -\sqrt{\mu}\\ \lambda_2 = -1 \end{cases}$
\begin{align*}
\lambda_1 &= -j &\text{ para} && \mu &= -1\\
\lambda_1 &= 0 &\text{ para} && \mu &= 0\\
\lambda_1 &= -1 &\text{ para} && \mu &= 1
\end{align*}
\end{minipage}
\vrule{}
\begin{minipage}[t]{0.3\linewidth}
$(\overline{x_1},\overline{x_2}) = (-\sqrt{\mu}, 0)$
$\begin{cases} \lambda_1 = \sqrt{\mu}\\ \lambda_2 = -1\\ \end{cases}$
\begin{align*}
\lambda_1 &= j &\text{ para} && \mu &= -1\\
\lambda_1 &= 0 &\text{ para} && \mu &= 0\\
\lambda_1 &= 1 &\text{ para} && \mu &= 1
\end{align*}
\end{minipage}
\end{document}

Thank you very much for your time and answers.

• If you are new to LaTeX and you would like to align your equations, please have a look at these two excellent online tutorials: Aligning several equations and align and align*. Your previous question had an answer by @egreg and you commented that you want to learn, so… learn :-) Three columns are no problem using that answer, you just need more &’s. – Ruixi Zhang Aug 18 '18 at 21:34
• I thought that minipages would be easier to do because I would be able to treat every minipage as one normal page. I did use this method in two other parts of my document, but this is the fist time it is not horizontally aligning and I wasn't able to find out why. Also, thank you very much for the comments, I will look at the tutorials you mentioned. – Marschall Aug 18 '18 at 21:41
• 1. Your code doesn’t compile (Missing \begin{document}). 2. Even if corrected, it does not exhibit the problem you describe. 3. Just to try to understand what you want, does the situation improve if you add \noindent at the end of each line that contains \begin{minipage}[t]{0.3\linewidth}, or does it get worse? – GuM Aug 18 '18 at 21:51
• Sorry for the bad post =/ The \noident did correct it, thank you very much @GuM – Marschall Aug 18 '18 at 21:59

Your problem could come from many places. For example, there could be not enough space to fit all three minipage’s (one of which is the paragraph indent mentioned by @GuM), so the minipage’s were pushed around and led to misalignment. This is one of the drawbacks of using the minipage approach.

I would suggest using the align* environment to align your equations. Here is my attempt. Some explanations are provided in the code as % <-.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[brazil]{babel}
\usepackage{amsmath}
\usepackage{amssymb}% <- For `\mathbb{R}'
\begin{document}
\subsection{Estabilidade de cada ponto de equilíbrio em relação a~$\mu$}
\begin{align*}
&  (\overline{x_1},\overline{x_2}) = (0, 0)
&& (\overline{x_1},\overline{x_2}) = (\sqrt{\mu}, 0)
&& (\overline{x_1},\overline{x_2}) = (-\sqrt{\mu}, 0) \\% <- First line finished
&  \begin{cases}
\lambda_1 = 0 \\
\lambda_2 = -1
\end{cases}
&& \begin{cases}
\lambda_1 = -\sqrt{\mu} \\
\lambda_2 = -1
\end{cases}
&& \begin{cases}
\lambda_1 = \sqrt{\mu} \\
\lambda_2 = -1\\
\end{cases} \\% <- Second line finished
&  \begin{aligned}[t]% <- Aligned at the top in this final line
\lambda_1 & = 0 & \forall \mu & \in \mathbb{R}% <- I used this to denote the set of real numbers
\end{aligned}
&& \begin{aligned}[t]% <- Aligned at the top in this final line
\lambda_1 & = -j & \text{para} \quad \mu & = -1 \\
\lambda_1 & = 0  & \text{para} \quad \mu & = 0  \\
\lambda_1 & = -1 & \text{para} \quad \mu & = 1
\end{aligned}
&& \begin{aligned}[t]% <- Aligned at the top in this final line
\lambda_1 & = j & \text{para} \quad \mu & = -1 \\
\lambda_1 & = 0 & \text{para} \quad \mu & = 0  \\
\lambda_1 & = 1 & \text{para} \quad \mu & = 1
\end{aligned}% <- Third line finished
\end{align*}
\end{document} Here is a code, without minipages, only alignat*{3}, some amsmath environments and the eqparbox package. I took the liberty to use \overline only with x, not the indices, as I think it looks better:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[TS1,T1]{fontenc}
\usepackage[showframe]{geometry}
\usepackage{multicol}
\usepackage{eqparbox}
\newcommand{\eqmathbox}[M]{\eqmakebox[#1]{$\displaystyle#2$}}
\usepackage{mathtools}
%\newcommand{\dydx}{\frac{d^{#1}{#2}}{d{#3}^{#4}}}
\usepackage{esdiff}

\begin{document}
\setcounter{section}{3}
\setcounter{subsection}{1}
\subsection{Estabilidade de cada ponto de equilíbrio em relação a \boldmath$\mu$}

\begin{alignat*}{3}
& \eqmathbox[L]{\begin{gathered}
(\overline{x}_1,\overline{x}_2) = (0, 0) \\
\begin{cases}
\lambda_1 = -2\sqrt{\mu }\\
\lambda_2 = -1
\end{cases}
\end{gathered}}
&\hspace{4em}
& \eqmathbox[C]{\begin{gathered}
(\overline{x}_1,\overline{x}_2) = (\sqrt{\mu }, 0) \\
\begin{cases}
\lambda_1 =0\\
\lambda_2 = -1
\end{cases}
\end{gathered}}
&\hspace{4em}
&\eqmathbox[R]{\begin{gathered}
(\overline{x}_1,\overline{x}_2) = (-\sqrt{\mu }, 0) \\
\begin{cases}
\lambda_1 = 2\sqrt{\mu }\\
\lambda_2 = -1
\end{cases}
\end{gathered}} \\[1ex]
& \eqmathbox[L]{\begin{aligned}[t]
\lambda_1 &= 0 &\forall &\mu \in \Re
\end{aligned}}
&
& \eqmathbox[C]{\begin{aligned}[t]
&\lambda_1, \lambda_2) = (-j2, -1), &x = - 1\\
&(\lambda_1, \lambda_2) = ( 0, -1), &x = 0\\
&(\lambda_1, \lambda_2) = (-2, -1), &x = 1
\end{aligned}}
&
& \eqmathbox[R]{\begin{aligned}[t]
&(\lambda_1, \lambda_2) = (j2, -1), &x = -&1\\
&(\lambda_1, \lambda_2) = ( 0, -1), &x = & 0\\
&(\lambda_1, \lambda_2) = (2, -1), &x = & 1
\end{aligned}}
\end{alignat*}

\end{document} 