Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 10 - Systems of Equations and Inequalities - Section 10.2 Systems of Linear Equations: Matrices - 10.2 Assess Your Understanding - Page 749: 23


$\left[\begin{array}{ccc|c} {1}&{7}&{-11}&{2}\\ {2}&{-5}&{6}&{-2}\\ {0}&{-9}&{16}&{2}\end{array}\right]$

Work Step by Step

The standard form of a linear equation can be expressed as: $a_{i1}x_{1}+a_{i2}x_{2}+...+a_{in}x_{n}=b_{i}$ where, the index $i$ indicates that it is the i-th equation of a system of equations. In order to write the augmented matrix $[A|B]$ of a system of equations in standard form, we must follow some important points: 1. To express a system in matrix form, we must extract the coefficients of the variables and constants. 2. Draw a vertical line to separate the coefficient entries from the constants (essentially replacing the equal signs). 3. The entries of the coefficient matrix $A=[a_{ij}]$ must be placed to the left of the line. 4. The constants of $B=[b_{i}]$ must be placed to the right of the line. We can write the system as an augmented matrix $[A|B]$ as follows: $\left\{\begin{array}{llll} 5x & -3y & +z & =-2\\ 2x & -5y & +6z & =-2\\ -4x & +y & +4z & =6 \end{array}\right. \rightarrow\left[\begin{array}{rrr|r} {5}&{-3}&{1}&{-2}\\ {2}&{-5}&{6}&{-2}\\ {-4}&{1}&{4}&{6}\end{array}\right]$ We perform the row operation as: $R_{1}=-2r_{2}+r_{1}$ $R_{3}=2r_{2}+r_{3}$ $=\left[\begin{array}{ccc|c} {(-2)(2)+5} &{(-2)(-5)-3}&{(-2)(6)+1}&{(-2)(-2)-2}\\ {2} &{-5} &{6} &{-2}\\ {2(2)+(-4)}&{(2)(-5)+1}&{2(6)+4}&{(2)(-2)+6}\end{array}\right]$ $=\left[\begin{array}{ccc|c} {1}&{7}&{-11}&{2}\\ {2}&{-5}&{6}&{-2}\\ {0}&{-9}&{16}&{2}\end{array}\right]$
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