1.2.1: Exercises 1.2
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In Exercises \(\PageIndex{1}\) - \(\PageIndex{4}\), convert the given system of linear equations into an augmented matrix.
\(\begin{array}{ccccccc} 3x&+&4y&+&5z&=&7\\ -x&+&y&-&3z&=&1\\ 2x&-&2y&+&3z&=&5\\ \end{array}\)
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\(\left[\begin{array}{cccc} 3&4&5&7\\-1&1&-3&1\\2&-2&3&5\\ \end{array}\right]\)
\(\begin{array}{ccccccc} 2x&+&5y&-&6z&=&2\\ 9x&&&-&8z&=&10\\ -2x&+&4y&+&z&=&-7\\ \end{array}\)
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\(\left[\begin{array}{cccc} 2&5&-6&2\\9&0&-8&10\\-2&4&1&-7\\ \end{array}\right]\)
\(\begin{array}{rl} x_1 +3x_2-4x_3 + 5x_4 =&17 \\ -x_1+4x_3+8x_4 =&1\\ 2x_1+3x_2+4x_3+5x_4=&6 \end{array}\)
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\(\left[\begin{array}{ccccc} 1&3&-4&5&17\\-1&0&4&8&1\\ 2&3&4&5&6\end{array}\right] \)
\(\begin{array}{rl} 3x_1 -2x_2=&4 \\ 2x_1 =&3\\ -x_1+9x_2=&8\\ 5x_1-7x_2=&13\\ \end{array}\)
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\(\left[\begin{array}{ccc} 3&-2&4\\ 2&0&3\\-1&9&8\\5&-7&13\\ \end{array}\right]\)
In Exercises \(\PageIndex{5}\) - \(\PageIndex{9}\), convert the given augmented matrix into a system of linear equations. Use the variables \(x_{1},\: x_{2}\), etc.
\(\left[\begin{array}{ccc} 1&2&3\\ -1&3&9\\ \end{array}\right]\)
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\(\begin{array}{rl} x_1+2x_2=&3\\ -x_1+3x_2=&9\\ \end{array}\)
\(\left[\begin{array}{ccc} -3&4&7\\ 0&1&-2\\ \end{array}\right]\)
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\(\begin{array}{rl} -3x_1+4x_2=&7\\ x_2=&-2\\ \end{array}\)
\(\left[\begin{array}{ccccc} 1&1&-1&-1&2\\ 2&1&3&5&7\\ \end{array}\right]\)
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\(\begin{array}{rl} x_1+x_2-x_3-x_4=&2\\ 2x_1+x_2+3x_3+5x_4=&7\\ \end{array}\)
\(\left[\begin{array}{ccccc} 1&0&0&0&2\\ 0&1&0&0&-1\\ 0&0&1&0&5\\ 0&0&0&1&3 \end{array}\right]\)
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\(\begin{array}{rl} x_1=&2\\ x_2=&-1\\ x_3=&5\\ x_4=&3\\ \end{array}\)
\(\left[\begin{array}{cccccc} 1&0&1&0&7&2\\ 0&1&3&2&0&5\\ \end{array}\right]\)
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\(\begin{array}{rl} x_1+x_3+7x_5=&2\\ x_2+3x_3+2x_4=&5\\ \end{array}\)
In Exercises \(\PageIndex{10}\) - \(\PageIndex{15}\), perform the given row operations on \(A\), where
\[A=\left[\begin{array}{ccc}{2}&{-1}&{7}\\{0}&{4}&{-2}\\{5}&{0}&{3}\end{array}\right]\nonumber \]
\(-1R_1\rightarrow R_1\)
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\(\left[\begin{array}{ccc} -2&1&-7\\0&4&-2\\5&0&3\\ \end{array}\right]\)
\(R_2\leftrightarrow R_3\)
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\(\left[\begin{array}{ccc} 2&-1&7\\5&0&3\\0&4&-2\\ \end{array}\right]\)
\(R_1+R_2\rightarrow R_2\)
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\(\left[\begin{array}{ccc} 2&-1&7\\2&3&5\\5&0&3\\ \end{array}\right]\)
\(2R_2+R_3\rightarrow R_3\)
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\(\left[\begin{array}{ccc} 2&-1&7\\0&4&-2\\5&8&-1\\ \end{array}\right]\)
\(\frac12R_2\rightarrow R_2\)
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\(\left[\begin{array}{ccc} 2&-1&7\\0&2&-1\\5&0&3\\ \end{array}\right]\)
\(-\frac52R_1+R_3\rightarrow R_3\)
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\(\left[\begin{array}{ccc} 2&-1&7\\0&4&-2\\0&5/2&-29/2\\ \end{array}\right]\)
A matrix \(A\) is given below. In Exercises \(\PageIndex{16}\) - \(\PageIndex{20}\), a matrix \(B\) is given. Give the row operation that transforms \(A\) into \(B\).
\[A=\left[\begin{array}{ccc}{1}&{1}&{1}\\{1}&{0}&{1}\\{1}&{2}&{3}\end{array}\right]\nonumber \]
\(B = \left[\begin{array}{ccc}1&1&1\\2&0&2\\1&2&3\\ \end{array}\right]\)
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\(2R_2\rightarrow R_2\)
\(B = \left[\begin{array}{ccc}1&1&1\\2&1&2\\1&2&3\\ \end{array}\right]\)
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\(R_1+R_2\rightarrow R_2\)
\(B = \left[\begin{array}{ccc}3&5&7\\1&0&1\\1&2&3\\ \end{array}\right]\)
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\(2R_3+R_1\rightarrow R_1\)
\(B= \left[\begin{array}{ccc}1&0&1\\1&1&1\\1&2&3\\ \end{array}\right]\)
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\(R_1\leftrightarrow R_2\)
\(B= \left[\begin{array}{ccc}1&1&1\\1&0&1\\0&2&2\\ \end{array}\right]\)
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\(-R_2+R_3\leftrightarrow R_3\)
In Exercises \(\PageIndex{21}\) - \(\PageIndex{26}\), rewrite the system of equations in matrix form. Find the solution to the linear system by simultaneously manipulating the equations and the matrix.
\(\begin{array}{ccccc} x&+&y&=&3\\ 2x&-&3y&=&1\\ \end{array}\)
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\(x=2,y=1\)
\(\begin{array}{ccccc} 2x&+&4y&=&10\\ -x&+&y&=&4\\ \end{array}\)
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\(x=-1,y=3\)
\(\begin{array}{ccccc} -2x&+&3y&=&2\\ -x&+&y&=&1\\ \end{array}\)
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\(x=-1,y=0\)
\(\begin{array}{ccccccc} 2x&+&3y&=&2\\ -2x&+&6y&=&1\\ \end{array}\)
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\(x=\frac12,y=\frac13\)
\(\begin{array}{ccccccc} -5x_1&&&+&2x_3&=&14\\ &&x_2&&&=&1\\ -3x_1&&&+&x_3&=&8\\ \end{array}\)
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\(x_1=-2,x_2=1,x_3=2\)
\(\begin{array}{ccccccc} &-&5x_2&+&2x_3&=&-11\\ x_1&&&+&2x_3&=&15\\ &-&3x_2&+&x_3&=&-8\\ \end{array}\)
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\(x_1=1,x_2=5,x_3=7\)