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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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