6.6E: Exercises
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Practice Makes Perfect
Write the Augmented Matrix for a System of Equations
In the following exercises, write each system of linear equations as an augmented matrix.
ⓐ \(\left\{ \begin{array} {l} 3x−y=−1\\ 2y=2x+5\end{array} \right.\)
ⓑ \(\left\{ \begin{array} {l} 4x+3y=−2\\ x−2y−3z=7 \\ 2x−y+2z=−6 \end{array} \right.\)
ⓐ \(\left\{ \begin{array} {l} 2x+4y=−5\\ 3x−2y=2\end{array} \right.\)
ⓑ \(\left\{ \begin{array} {l} 3x−2y−z=−2\\ −2x+y=5 \\ 5x+4y+z=−1 \end{array} \right.\)
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ⓐ \(\left[ \begin{matrix} 2 &4 &−5 \\ 3 &−2 &2 \end{matrix} \right]\)
ⓑ \(\left[ \begin{matrix} 3 &−2 &−1 &−2 \\ −2 &1 &0 &5 \\ 5 &4 &1 &−1 \end{matrix} \right]\)
ⓐ \(\left\{ \begin{array} {l} 3x−y=−4 \\ 2x=y+2 \end{array} \right.\)
ⓑ \(\left\{ \begin{array} {l} x−3y−4z=−2 \\ 4x+2y+2z=5 \\ 2x−5y+7z=−8 \end{array} \right.\)
ⓐ \(\left\{ \begin{array} {l} 2x−5y=−3 \\ 4x=3y−1 \end{array} \right.\)
ⓑ \(\left\{ \begin{array} {l} 4x+3y−2z=−3 \\ −2x+y−3z=4 \\ −x−4y+5z=−2 \end{array} \right.\)
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ⓐ \(\left[ \begin{matrix} 2 &−5 &−3 \\ 4 &−3 &−1 \end{matrix} \right]\)
ⓑ \(\left[ \begin{matrix} 4 &3 &−2 &−3 \\ −2 &1 &−3 &4 \\ −1 &−4 &5 &−2 \end{matrix} \right]\)
Write the system of equations that corresponds to the augmented matrix.
\(\left[ \begin{array} {cc|c} 2 &−1 &4 \\ 1 &−3 &2 \end{array} \right]\)
\(\left[ \begin{array} {cc|c} 2 &−4 &-2 \\ 3 &−3 &-1 \end{array} \right]\)
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\(\left\{ \begin{array} {l} 2x−4y=−2 \\ 3x−3y=−1 \end{array} \right.\)
\(\left[ \begin{array} {ccc|c} 1 &0 &−3 &-1 \\ 1 &−2 &0 &-2 \\ 0 &−1 &2 &3 \end{array} \right]\)
\(\left[ \begin{array} {ccc|c} 2 &−2 &0 &-1 \\ 0 &2 &−1 &2 \\ 3 &0 &−1 &-2 \end{array} \right]\)
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\(\left\{ \begin{array} {l} 2x−2y=−1 \\ 2y−z=2 \\ 3x−z=−2 \end{array} \right.\)
Use Row Operations on a Matrix
In the following exercises, perform the indicated operations on the augmented matrices.
\(\left[ \begin{array} {cc|c} 6 &−4 &3 \\ 3 &−2 &1 \end{array} \right]\)
ⓐ Interchange rows 1 and 2
ⓑ Multiply row 2 by 3
ⓒ Multiply row 2 by \(−2\) and add row 1 to it.
\(\left[ \begin{array} {cc|c} 4 &−6 &-3 \\ 3 &2 &1 \end{array} \right]\)
ⓐ Interchange rows 1 and 2
ⓑ Multiply row 1 by 4
ⓒ Multiply row 2 by 3 and add row 1 to it.
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ⓐ \(\left[ \begin{matrix} 3 &2 &1 \\ 4 &−6 &−3 \end{matrix} \right]\)
ⓑ \(\left[ \begin{matrix} 12 &8 &4 \\ 4 &−6 &−3 \end{matrix} \right]\)
ⓒ \(\left[ \begin{matrix} 12 &8 &4 \\ 24 &−10 &−5 \end{matrix} \right]\)
\(\left[ \begin{array} {ccc|c} 4 &−12 &−8 &16 \\ 4 &−2 &−3 &-1 \\ −6 &2 &−1 &-1 \end{array} \right]\)
\(\left[ \begin{array} {ccc|c} 6 &−5 &2 &3 \\ 2 &1 &−4 &5 \\ 3 &−3 &1 &-1 \end{array} \right]\)
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ⓐ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ 3 &−3 &1 &−1 \end{matrix} \right]\)
ⓑ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ 3 &−3 &1 &−1 \end{matrix} \right]\)
ⓒ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ −4 &7 &−6 &7 \end{matrix} \right]\)
Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: \(\left[ \begin{array} {cc|c} 1 &2 &5 \\ −3 &−4 &-1 \end{array} \right]\)
Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: \(\left[ \begin{array} {ccc|c} 1 &−2 &3 &-4 \\ 3 &−1 &−2 &5 \\ 2 &−3 &−4 &1 \end{array} \right]\)
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\(\left[ \begin{matrix} 1 &−2 &3 &−4 \\ 0 &5 &−11 &17 \\ 0 &1 &−10 &7 \end{matrix} \right]\)
Solve Systems of Equations Using Matrices
In the following exercises, solve each system of equations using a matrix.
\(\left\{ \begin{array} {l} 2x+y=2 \\ x−y=−2 \end{array} \right.\)
\(\left\{ \begin{array} {l} 3x+y=2 \\ x−y=2 \end{array} \right.\)
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\((1,−1)\)
\(\left\{ \begin{array} {l} −x+2y=−2 \\ x+y=−4 \end{array} \right.\)
\(\left\{ \begin{array} {l} −2x+3y=3 \\ x+3y=12 \end{array} \right.\)
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\((3,3)\)
In the following exercises, solve each system of equations using a matrix.
\(\left\{ \begin{array} {l} 2x−3y+z=19 \\ −3x+y−2z=−1 \\ 5x+y+z=0 \end{array} \right.\)
\(\left\{ \begin{array} {l} 2x−y+3z=−3 \\ −x+2y−z=10 \\ x+y+z=5 \end{array} \right.\)
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\((−2,5,2)\)
\(\left\{ \begin{array} {l} 2x−6y+z=3 \\ 3x+2y−3z=2 \\ 2x+3y−2z=3 \end{array} \right.\)
\(\left\{ \begin{array} {l} 4x−3y+z=7 \\ 2x−5y−4z=3 \\ 3x−2y−2z=−7 \end{array} \right.\)
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\((−3,−5,4)\)
\(\left\{ \begin{array} {l} x+2z=0 \\ 4y+3z=−2 \\ 2x−5y=3 \end{array} \right.\)
\(\left\{ \begin{array} {l} 2x+5y=4 \\ 3y−z=3 \\ 4x+3z=−3 \end{array} \right.\)
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\((−3,2,3)\)
\(\left\{ \begin{array} {l} 2y+3z=−1 \\ 5x+3y=−6 \\ 7x+z=1 \end{array} \right.\)
\(\left\{ \begin{array} {l} 3x−z=−3 \\ 5y+2z=−6 \\ 4x+3y=−8 \end{array} \right.\)
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\((−2,0,−3)\)
\(\left\{ \begin{array} {l} 2x+3y+z=1 \\ 2x+y+z=9 \\ 3x+4y+2z=20 \end{array} \right.\)
\(\left\{ \begin{array} {l} x+2y+6z=5 \\ −x+y−2z=3 \\ x−4y−2z=1 \end{array} \right.\)
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no solution
\(\left\{ \begin{array} {l} x+2y−3z=−1 \\ x−3y+z=1 \\ 2x−y−2z=2 \end{array} \right.\)
\(\left\{ \begin{array} {l} 4x−3y+2z=0 \\ −2x+3y−7z=1 \\ 2x−2y+3z=6 \end{array} \right.\)
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no solution
\(\left\{ \begin{array} {l} x−y+2z=−4 \\ 2x+y+3z=2 \\ −3x+3y−6z=12 \end{array} \right.\)
\(\left\{ \begin{array} {l} −x−3y+2z=14 \\ −x+2y−3z=−4 \\ 3x+y−2z=6 \end{array} \right.\)
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infinitely many solutions \((x,y,z)\) where \(x=12z+4;\space y=12z−6;\space z\) is any real number
\(\left\{ \begin{array} {l} x+y−3z=−1 \\ y−z=0 \\ −x+2y=1 \end{array} \right.\)
\(\left\{ \begin{array} {l} x+2y+z=4 \\ x+y−2z=3 \\ −2x−3y+z=−7 \end{array} \right.\)
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infinitely many solutions \((x,y,z)\) where \(x=5z+2;\space y=−3z+1;\space z\) is any real number
Writing Exercises
Solve the system of equations \(\left\{ \begin{array} {l} x+y=10 \\ x−y=6\end{array} \right.\) ⓐ by graphing and ⓑ by substitution. ⓒ Which method do you prefer? Why?
Solve the system of equations \(\left\{ \begin{array} {l} 3x+y=1 \\ 2x=y−8 \end{array} \right.\) by substitution and explain all your steps in words.
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Answers will vary.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?