Systems of equations (§12.1)

If you're solving several equations in several variables and you want a single solution that satisfies all of the equations at once, then you're solving a system of equations.

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.

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