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Unformatted text preview: tem is in triangular form.12 An example of a more complicated system in triangular form is x1 − 4x3 + x4 − x6 = 6 x2 + 2 x3 = 1
x4 + 3 x5 − x6 = 8 x5 + 9x6 = 10
Our goal henceforth will be to transform a given system of linear equations into triangular form
using the moves in Theorem 8.1.
Example 8.1.2. Use Theorem 8.1 to put the following systems into triangular form and then solve
the system if possible. Classify each system as consistent independent, consistent dependent, or
inconsistent. 3x − y + z = 3 2x + 3y − z = 1 3x1 + x2 + x4 = 6
2x − 4y + 3z = 16
10x − z = 2
2x1 + x2 − x3 = 4
3. x−y+z = 5
4x − 9y + 2z = 5
x2 − 3x3 − 2x4 = 0
1. For deﬁnitiveness, we label the topmost equation in the system E 1, the equation beneath that
E 2, and so forth. We now attempt to put the system in triangular form using an algorithm
known as Gaussian Elimination. What this means is that, starting with x, we transform
the system so that conditions 2 and 3 in Deﬁnition 8.3 are satisﬁed. Then we...
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