What are Linear Functions?

All of the equations in a linear program must, by definition, be linear. A linear function has the following form:

a0 + a1 x1 + a2 x2 + a3 x3 + . . . + an xn = 0

In general, the a's are called the coefficients of the equation; they are also sometimes called parameters. The important thing to know about the coefficients is that they are fixed values, based on the underlying nature of the problem being solved. The x's are called the variables of the equation; they are allowed to take on a range of values within the limits defined by the constraints. Note that it is not necessary to always use x's to represent variables; any label could be used, and more descriptive labels are often more useful.

Linear equations and inequalities are often written using summation notation, which makes it possible to write an equation in a much more compact form. The linear equation above, for example, can be written as follows:

Note that the letter i is an index, or counter, that starts in this case at 1 and runs to n. There is a term in the sum for each value of the index. Just as a variable does not have to be specified with a letter x, the index does not have to be a letter i. Summation notation will be used a lot in the rest of this chapter and in all of the remaining chapters. You will need to become adept at interpreting it.

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