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Compare these approaches:

Approach 1 (of yours).

\begin{equation*}
\left \{
\begin{aligned}
u_t + b \cdot D u + c u &= 0 & \text{ inside } & \Omega \\
u &= g_1 + g_2 + g_3 & \text{ on } & \partial\Omega.
\end{aligned}
\right.
\end{equation*}


Approach 2.

\begin{equation*}
\left ...

Thank you very much. But I want the equations to be aligned not only in the "=" sign but also with respect to "in" and "on". If you write a longer expression instead of "g" in the second equation, you'll see that there's no alignment with respect to "in" and "on".

I tried substituting "\quad" with ...

Hello! How to make equations aligned as in the problem 1 in the attached file?