Intermediate Algebra for College Students (7th Edition)

Published by Pearson
ISBN 10: 0-13417-894-7
ISBN 13: 978-0-13417-894-3

Chapter 3 - Section 3.4 - Matrix Solutions to Linear Systems - Exercise Set - Page 229: 27

Answer

$\{(3,-1,-1)\}$.

Work Step by Step

The given system of equations is $x+3y=0$ $x+y+z=1$ $3x-y-z=11$ The augmented matrix is $\Rightarrow \left[\begin{array}{ccc|c} 1 & 3 & 0& 0\\ 1 &1 & 1& 1 \\ 3&-1&-1&11 \end{array}\right]$ Perform $R_2\rightarrow R_2-R_1$ and $R_3\rightarrow R_3-3( R_1)$. $\Rightarrow \left[\begin{array}{ccc|c} 1 & 3 & 0& 0\\ 1-1 &1 -3& 1-0& 1-0 \\ 3-3(1)&-1-3(3)&-1-3(0)&11-3(0) \end{array}\right]$ Simplify. $\Rightarrow \left[\begin{array}{ccc|c} 1 & 3 & 0& 0\\ 0 &-2& 1& 1 \\ 0&-10&-1&11 \end{array}\right]$ Perform $R_2\Rightarrow R_2/(-2)$ Change the sign of the second row and swap the 3rd and 2nd row. $\Rightarrow \left[\begin{array}{ccc|c} 1 & 3 & 0& 0\\ 0/(-2) &-2/(-2)& 1/(-2)& 1/(-2) \\ 0&-10&-1&11 \end{array}\right]$ Simplify. $\Rightarrow \left[\begin{array}{ccc|c} 1 & 3 & 0& 0\\ 0 &1& -1/2& -1/2 \\ 0&-10&-1&11 \end{array}\right]$ Perform $R_1\rightarrow R_1-3\times R_2$ and $R_3\rightarrow R_3+10 R_2$. $\Rightarrow \left[\begin{array}{ccc|c} 1-3(0) & 3-3(1) & 0-3(-1/2)& 0-3(-1/2)\\ 0 &1& -1/2& -1/2 \\ 0+10(0)&-10+10(1)&-1+10(-1/2)&11+10(-1/2) \end{array}\right]$ Simplify. $\Rightarrow \left[\begin{array}{ccc|c} 1 & 0 & 3/2& 3/2\\ 0 &1& -1/2& -1/2 \\ 0&0&-6&6 \end{array}\right]$ Perform $R_3\rightarrow \frac{R_3}{-6}$. $\Rightarrow \left[\begin{array}{ccc|c} 1 & 0 & 3/2& 3/2\\ 0 &1& -1/2& -1/2 \\ 0/(-6) & 0 /(-6)& -6/(-6)& 6/(-6) \end{array}\right]$ Simplify. $\Rightarrow \left[\begin{array}{ccc|c} 1 & 0 & 3/2& 3/2\\ 0 &1& -1/2& -1/2 \\ 0 & 0& 1& -1 \end{array}\right]$ Perform $R_1\rightarrow R_1-(3/2)R_3$ and $R_2\rightarrow R_2+(1/2) R_3$. $\Rightarrow \left[\begin{array}{ccc|c} 1-(3/2)0 & 0-(3/2)0 & 3/2-(3/2)1& 3/2-(3/2)(-1)\\ 0+(1/2)0 &1+(1/2)0& -1/2+(1/2)1& -1/2+(1/2)(-1) \\ 0 & 0& 1& -1 \end{array}\right]$ Simplify. $\Rightarrow \left[\begin{array}{ccc|c} 1 & 0 & 0& 3\\ 0 &1& 0& -1 \\ 0 & 0& 1& -1 \end{array}\right]$ Use back substitution to solve the linear system. $\Rightarrow x=3$ and $\Rightarrow y=-1$. and $\Rightarrow z=-1$. The solution set is $\{(x,y,z)\}=\{(3,-1,-1)\}$.
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