Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 10 - Section 10.3 - Matrices and Systems of Linear Equations - 10.3 Exercises - Page 710: 52

Answer

$(7t-5,8t-4,t)$

Work Step by Step

Step 1. Establish the augmented matrix of the system and use the Gauss Eliminations method: $\begin{vmatrix} -4 & -1 & 36 & 24 \\ 1 & -2 & 9 & 3\\-2 & 1 & 6 & 6 \end{vmatrix} \begin{array}(R_2\leftrightarrow R_1 \\.\\.\\ \end{array}$ Step 2. exchange row2 with row 1: $\begin{vmatrix} 1 & -2 & 9 & 3 \\ -4 & -1 & 36 & 24\\-2 & 1 & 6 & 6 \end{vmatrix} \begin{array} . \\R_2+4R_1\to R_2\\R_3+2R_1\to R_3\end{array}$ Step 3. Do the operations given on the right side of the matrix. $\begin{vmatrix} 1 & -2 & 9 & 3 \\ 0 & -9 & 72 & 36\\0 & -3 & 24 & 12 \end{vmatrix} \begin{array} . \\R_2/(-9)\to R_2\\R_3/(-3)\to R_3\end{array}$ Step 4. Simplify the second and third rows: $\begin{vmatrix} 1 & -2 & 9 & 3 \\ 0 & 1 & -8 & -4\\0 & 1 & -8 & -4 \end{vmatrix}$ Step 5. Because the third row is the same as the second, we have dependent equations and unlimited solutions. Let $z=t$ and write the equations from step 4 as: $\begin{cases} x-2y+9t=3 \\ y-8t=-4 \end{cases} $ which gives the solutions as $x=7t-5,y=8t-4,z=t$
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