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Unformatted text preview: then solve for x. We formalize this pattern below for the most general systems of linear equations. Again, we use subscripted variables to describe the general case. The variable with the smallest subscript in a given equation is typically called the leading variable of that equation. Definition 8.3. A system of linear equations with variables x1 , x2 , . . . xn is said to be in triangular form provided all of the following conditions hold: 1. The subscripts of the variables in each equation are always increasing from left to right. 2. The leading variable in each equation has coefficient 1. 3. The subscript on the leading variable in a given equation is greater than the subscript on the leading variable in the equation above it. 4. Any equation without variablesa cannot be placed above an equation with variables. a necessarily an identity or contradiction In our previous system, if make the obvious choices x = x1 , y = x2 , and z = x3 , we see that the 8.1 Systems of Linear Equations: Gaussian Elimination 457 sys...
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