Linear systems in Latex

Define a single equation

To represent and inline equation like this: $a + b + c = 10$, use the following syntax:

$$a + b + c = 10$$

But if we want to define an equation in a linear system like this:

\begin{equation}
a + b + c = 10
\end{equation}

More than one equation

In separate systems:

\begin{align}
a + b &= 10 \\
a + 2b &= 12
\end{align}

In the same system:

\begin{equation}
  \left{
    \begin{array}{rcl}
      a + b & = & 10 \\
      a + 2b & = & 12
    \end{array}
  \right
  \label{eq:system}
\end{equation}

The & is used for alignment.

In matrix form

This system of equations can be expressed in matrix form as follows:

\begin{bmatrix}
  1 & 1 \\
  1 & 2
\end{bmatrix}

\begin{bmatrix}
  a \\
  b
\end{bmatrix}

=

\begin{bmatrix}
  10 \\
  12
\end{bmatrix}