Generally, you should have the same number of equations as variables, and this should remain true as you go through the process of solving the system. So if you substitute one equation into another, add equations together, etc, then your new equation should always replace one of your old equations, so that the number of equations doesn't change. The exceptions are for dependent systems of equations, where eventually one of your equations becomes always true or always false. If it becomes always true, then you throw it out, and from then on, you have fewer equations. If it becomes always false, then you throw the whole system out; it is inconsistent and has no solutions. Otherwise, you keep the same number of equations and the same number of variables ―that is, even if some individual equations have fewer variables, the system as a whole should keep the same number of variables— until the system is solved.
Here's an example; check out the video:
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This web page and the files linked from it were written in 2016 by Toby Bartels, last edited on 2016 August 30. Toby reserves no legal rights to them. The linked files were produced using GIMP and recordMyDesktop and converted from Ogg Vorbis to other formats by online-convert.com.
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