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Unformatted text preview: d systems,7 but a consistent underdetermined system of linear equations is necessarily dependent.8 In order to move this section beyond a review of Intermediate Algebra, we now define what is meant by a linear equation in n variables. Definition 8.2. A linear equation in n variables, x1 , x2 , . . . , xn is an equation of the form a1 x1 + a2 x2 + . . . + an xn = c where a1 , a2 , . . . an and c are real numbers and at least one of a1 , a2 , . . . , an is nonzero. Instead of using more familiar variables like x, y , and even z and/or w in Definition 8.2, we use subscripts to distinguish the different variables. We have no idea how many variables may be involved, so we use numbers to distinguish them instead of letters. (There is an endless supply of distinct numbers.) As an example, the linear equation 3x1 − x2 = 4 represents the same relationship between the variables x1 and x2 as the equation 3x − y = 4 does between the variables x and y . In addition, just as we cannot combine the terms in the expression 3x − y , we cannot com...
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