I'm sure most all posters on this forum can solve systems of linear equations with multiple variables using the classical "substitution" method. Are there methods deeper within linear algebra that allows for deriving a solution faster rather than the tedious "solving and plugging in" technique?

For members that aren't too familiar with linear equations, they are merely equations with a maximum exponent of one, taking the form of a polynomial. The substituion method I described above is similar to such (using a rather elementary linear equation, there are much more difficult ones):

4 + 3x= 16

-4____-4

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3x= 12

/3__/3

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x= 4

{4}