Intermediate Algebra for College Students (7th Edition)

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

Chapter 11 - Cumulative Review Exercises - Page 872: 7

Answer

The solution set is$\underline{\left( 4,0,-5 \right)}$.

Work Step by Step

The given equation set is\[\begin{align} & 3x-2y+z=7 \\ & 2x+3y-z=13 \\ & x-y+2z=-6 \end{align}\]. Calculation: The provided equations are, \[3x-2y+z=7\] …(1) \[2x+3y-z=13\]…(2) \[x-y+2z=-6\]…(3) Simplify as follows: Step 1: Let’s eliminate$z$and make two equations in two variables, use the addition method in equation (1) and (2) to eliminate the variable$z$, \[3x-2y+z+\left( 2x+3y-z \right)=7+13\] \[5x+y=20\] …(4) And multiply by $2$ in equation (2), use the addition method in equation (2) and (3) to eliminate the variable$z$, \[4x+6y-2z+\left( x-y+2z \right)=26+\left( -6 \right)\] $5x+5y=20$ …(5) Step 2: Subtracting equation (5) from (4) to eliminate the variable$x$, $\begin{align} & 5x+y-\left( 5x+5y \right)=20-20 \\ & 5x+y-5x-5y=0 \\ & -4y=0 \\ & y=0 \end{align}$ Step 3: Put the value of $y$ in equation (4) and simplify as follows, $\begin{align} & 5x+\left( 0 \right)=20 \\ & 5x=20 \\ & x=4 \end{align}$ Step 4: Put the value of $x$ and $y$ in equation (1) and find out the value of$z$, simplify as follows: \[\begin{align} & 3x-2y+z=7 \\ & 3\left( 4 \right)-2\left( 0 \right)+z=7 \\ & 12+z=7 \\ & z=-5 \end{align}\] Therefore, these three equations have a solution of$x=4,\,\,y=0\text{ and }z=-5$.
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