Systems of Linear Equations

“Where shall I begin, please your Majesty?” he asked.“Begin at the beginning,” the King said, gravely, “and go on till you come to the end: then stop.” 

—Lewis Carroll: Alice’s Adventures in Wonderland, 1865

Before learning everything and anything, we might want to need the base in every of linear equations.

   The word trivial is derived from the Latin root tri (“three”) and the Latin word via (“road”). Thus, speaking literally, a triviality is a place where three roads meet. The same system of equations can be viewed in three different, yet equally important, ways—these will be our three roads, all leading to the same solution. 

   Recall that the general equation of a line, ax + by = c, and that the general equation of a plane, ax  by  cz  d. Equations of this form are called linear equations.

Now, what is linear equation?

Are these examples of linear equations?

YES!

It is important to note that, although in these examples (and in most applications) the coefficients and constant terms are real numbers

Are these examples of linear equations?

NO!

Thus, linear equations do not contain products, reciprocals, or other functions of the variables; the variables occur only to the first power and are multiplied only by constants.