Is there a difference between "Linear System of Equations" and "System of linear Equations"?

Question

We are translating German math videos to English and stumbled over the fact that there are two translations of "Lineare Gleichungssysteme":

• Linear System of Equations
• System of linear Equations (e.g. Wikipedia)

From my point of view those describe two different things. A "linear system" and second a "… system" (which does not have to be linear).

But it seems to be used for the same thing. Why? And is it correct?

PS: Here is the translated video with the (recent) title "Linear System of Equations": https://www.youtube.com/watch?v=SPu8dHq_9VA

Answer 1

The two terms are interchangeable and mean the same thing. The term "linear system of equations" should be broken up as "(linear) (system of equations)", meaning that the system of equations is linear, i.e. that it involves equations which are linear. This is the same as a "system of linear equations", of course. You can't have a "linear system of equations" with nonlinear equations---the linearity of the equations exactly determines that of the system, they are not separate things, so the distinction is not meaningful.

Answer 2

Another synonym is "Simultaneous linear equations", which would be equivalent with "Linear simultaneous equations". So, linearity describes the equations at the end. However, I would prefer "linear equations" put together for more clarity.