# Chapter 8 - Section 8.2 - Systems of Linear Equations: Matrices - 8.2 Assess Your Understanding - Page 570: 21

The system represented by the augmented matrix is $\left\{\begin{array}{llll} x & -3y & +4z & =3\\ 3x & -5y & +6z & =6\\ -5x & +3y & +4z & =6 \end{array}\right.$ Performing the row operations, $\rightarrow\left[\begin{array}{ccc|c} {1}&{-3}&{4}&{3}\\ {0}&{4}&{-6}&{-3}\\ {0}&{-12}&{24}&{21}\end{array}\right]$

#### Work Step by Step

The system represented by the augmented matrix is $\left\{\begin{array}{llll} x & -3y & +4z & =3\\ 3x & -5y & +6z & =6\\ -5x & +3y & +4z & =6 \end{array}\right.$ Performing the row operations,$\begin{array}{l}{R_{2}=-3 r_{1}+r_{2}} \\ {R_{3}=5 r_{1}+r_{3}}\end{array}$ $\left[\begin{array}{ccc|c} {1}&{-3}&{4}&{3}\\ {3}&{-5}&{6}&{6}\\ {-5}&{3}&{4}&{6}\end{array}\right]\rightarrow\left[\begin{array}{ccc|c} {1} &{-3} &{4} &{3}\\ {-3(1)+3}&{-3(-3)-5}&{-3(4)+6}&{-3(3)+6}\\ {5(1)-5}&{5(-3)+3}&{5(4)+4}&{5(3)+6}\end{array}\right]$ $\rightarrow\left[\begin{array}{ccc|c} {1}&{-3}&{4}&{3}\\ {0}&{4}&{-6}&{-3}\\ {0}&{-12}&{24}&{21}\end{array}\right]$

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