**Unformatted text preview: **e 8.2.1. Use an augmented matrix to transform
into triangular form. Solve the system. 3x − y + z =
x + 2y − z = 2x + 3y − 4z = the following system of linear equations
8
4
10 Solution. We ﬁrst encode the system into an augmented matrix. 3 −1
18 3x − y + z = 8
Encode into the matrix
x + 2y − z = 4
2 −1 4 −− − − − − −→ 1
−−−−−−− 2x + 3y − 4z = 10
2
3 −4 10
Thinking back to Gaussian Elimination at an equations level, our ﬁrst order of business is to get x
in E 1 with a coeﬃcient of 1. At the matrix level, this means getting a leading 1 in R1. This is in
accordance with the ﬁrst criteria in Deﬁnition 8.4. To that end, we interchange R1 and R2. 3 −1
18
1
2 −1 4
Switch R1 and R2
1
2 −1 4 − − − − − −
1 8
− − − − − → 3 −1
2
3 −4 10
2
3 −4 10 8.2 Systems of Linear Equations: Augmented Matrices 469 Our next step is to eliminate the x’s from E 2 and E 3. From a matrix standpoint, this means we
need 0’s below the leading 1 in R1. This guarantees the leading 1 in R2 will be to the right of the
leading 1 in R1 in accordance with the second requirement of Deﬁnition 8.4. 4
1
2 −1 4
1
2 −1
Replace R2 with −3R1 + R2 3 −1
1 8 −−...

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