Linear Algebra Examples

$2 a + b - d - 2 g + 2 h + j + 5 k = 21$ , $a + b - 3 d + g + h + j + 2 k = - 5$ , $a + 2 b - 8 d + 5 g + h + j - 6 k = - 15$ , $3 a + 3 b - 9 d + 3 g + 6 h + 5 j + 2 k = - 24$ , $- 2 a - b + d + 2 g + h + j - 9 k = - 30$

Step 1

Write the system of equations in matrix form.

$[\begin{array}{c} 2 & 1 & - 1 & - 2 & 2 & 1 & 5 & 21 \\ 1 & 1 & - 3 & 1 & 1 & 1 & 2 & - 5 \\ 1 & 2 & - 8 & 5 & 1 & 1 & - 6 & - 15 \\ 3 & 3 & - 9 & 3 & 6 & 5 & 2 & - 24 \\ - 2 & - 1 & 1 & 2 & 1 & 1 & - 9 & - 30 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Step 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{1}{2} & \frac{- 1}{2} & \frac{- 2}{2} & \frac{2}{2} & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 1 & 1 & - 3 & 1 & 1 & 1 & 2 & - 5 \\ 1 & 2 & - 8 & 5 & 1 & 1 & - 6 & - 15 \\ 3 & 3 & - 9 & 3 & 6 & 5 & 2 & - 24 \\ - 2 & - 1 & 1 & 2 & 1 & 1 & - 9 & - 30 \end{array}]$

Step 2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 1 & 1 & - 3 & 1 & 1 & 1 & 2 & - 5 \\ 1 & 2 & - 8 & 5 & 1 & 1 & - 6 & - 15 \\ 3 & 3 & - 9 & 3 & 6 & 5 & 2 & - 24 \\ - 2 & - 1 & 1 & 2 & 1 & 1 & - 9 & - 30 \end{array}]$

Step 2.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 1 - 1 & 1 - \frac{1}{2} & - 3 + \frac{1}{2} & 1 + 1 & 1 - 1 & 1 - \frac{1}{2} & 2 - \frac{5}{2} & - 5 - \frac{21}{2} \\ 1 & 2 & - 8 & 5 & 1 & 1 & - 6 & - 15 \\ 3 & 3 & - 9 & 3 & 6 & 5 & 2 & - 24 \\ - 2 & - 1 & 1 & 2 & 1 & 1 & - 9 & - 30 \end{array}]$

Step 2.2.2

Simplify $R_{2}$ .

Step 2.3

Perform the row operation $R_{3} = R_{3} - R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.3.1

Perform the row operation $R_{3} = R_{3} - R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & \frac{1}{2} & - \frac{5}{2} & 2 & 0 & \frac{1}{2} & - \frac{1}{2} & - \frac{31}{2} \\ 1 - 1 & 2 - \frac{1}{2} & - 8 + \frac{1}{2} & 5 + 1 & 1 - 1 & 1 - \frac{1}{2} & - 6 - \frac{5}{2} & - 15 - \frac{21}{2} \\ 3 & 3 & - 9 & 3 & 6 & 5 & 2 & - 24 \\ - 2 & - 1 & 1 & 2 & 1 & 1 & - 9 & - 30 \end{array}]$

Step 2.3.2

Simplify $R_{3}$ .

Step 2.4

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 2.4.1

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & \frac{1}{2} & - \frac{5}{2} & 2 & 0 & \frac{1}{2} & - \frac{1}{2} & - \frac{31}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 0 & \frac{1}{2} & - \frac{17}{2} & - \frac{51}{2} \\ 3 - 3 \cdot 1 & 3 - 3 (\frac{1}{2}) & - 9 - 3 (- \frac{1}{2}) & 3 - 3 \cdot - 1 & 6 - 3 \cdot 1 & 5 - 3 (\frac{1}{2}) & 2 - 3 (\frac{5}{2}) & - 24 - 3 (\frac{21}{2}) \\ - 2 & - 1 & 1 & 2 & 1 & 1 & - 9 & - 30 \end{array}]$

Step 2.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & \frac{1}{2} & - \frac{5}{2} & 2 & 0 & \frac{1}{2} & - \frac{1}{2} & - \frac{31}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 0 & \frac{1}{2} & - \frac{17}{2} & - \frac{51}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 3 & \frac{7}{2} & - \frac{11}{2} & - \frac{111}{2} \\ - 2 & - 1 & 1 & 2 & 1 & 1 & - 9 & - 30 \end{array}]$

Step 2.5

Perform the row operation $R_{5} = R_{5} + 2 R_{1}$ to make the entry at $5, 1$ a $0$ .

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Step 2.5.1

Perform the row operation $R_{5} = R_{5} + 2 R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & \frac{1}{2} & - \frac{5}{2} & 2 & 0 & \frac{1}{2} & - \frac{1}{2} & - \frac{31}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 0 & \frac{1}{2} & - \frac{17}{2} & - \frac{51}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 3 & \frac{7}{2} & - \frac{11}{2} & - \frac{111}{2} \\ - 2 + 2 \cdot 1 & - 1 + 2 (\frac{1}{2}) & 1 + 2 (- \frac{1}{2}) & 2 + 2 \cdot - 1 & 1 + 2 \cdot 1 & 1 + 2 (\frac{1}{2}) & - 9 + 2 (\frac{5}{2}) & - 30 + 2 (\frac{21}{2}) \end{array}]$

Step 2.5.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & \frac{1}{2} & - \frac{5}{2} & 2 & 0 & \frac{1}{2} & - \frac{1}{2} & - \frac{31}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 0 & \frac{1}{2} & - \frac{17}{2} & - \frac{51}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 3 & \frac{7}{2} & - \frac{11}{2} & - \frac{111}{2} \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.6

Multiply each element of $R_{2}$ by $2$ to make the entry at $2, 2$ a $1$ .

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Step 2.6.1

Multiply each element of $R_{2}$ by $2$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 2 \cdot 0 & 2 (\frac{1}{2}) & 2 (- \frac{5}{2}) & 2 \cdot 2 & 2 \cdot 0 & 2 (\frac{1}{2}) & 2 (- \frac{1}{2}) & 2 (- \frac{31}{2}) \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 0 & \frac{1}{2} & - \frac{17}{2} & - \frac{51}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 3 & \frac{7}{2} & - \frac{11}{2} & - \frac{111}{2} \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.6.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 0 & \frac{1}{2} & - \frac{17}{2} & - \frac{51}{2} \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 3 & \frac{7}{2} & - \frac{11}{2} & - \frac{111}{2} \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.7

Perform the row operation $R_{3} = R_{3} - \frac{3}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 2.7.1

Perform the row operation $R_{3} = R_{3} - \frac{3}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & - \frac{15}{2} - \frac{3}{2} \cdot - 5 & 6 - \frac{3}{2} \cdot 4 & 0 - \frac{3}{2} \cdot 0 & \frac{1}{2} - \frac{3}{2} \cdot 1 & - \frac{17}{2} - \frac{3}{2} \cdot - 1 & - \frac{51}{2} - \frac{3}{2} \cdot - 31 \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 3 & \frac{7}{2} & - \frac{11}{2} & - \frac{111}{2} \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & 0 & 0 & 0 & 0 & - 1 & - 7 & 21 \\ 0 & \frac{3}{2} & - \frac{15}{2} & 6 & 3 & \frac{7}{2} & - \frac{11}{2} & - \frac{111}{2} \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.8

Perform the row operation $R_{4} = R_{4} - \frac{3}{2} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 2.8.1

Perform the row operation $R_{4} = R_{4} - \frac{3}{2} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & 0 & 0 & 0 & 0 & - 1 & - 7 & 21 \\ 0 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & - \frac{15}{2} - \frac{3}{2} \cdot - 5 & 6 - \frac{3}{2} \cdot 4 & 3 - \frac{3}{2} \cdot 0 & \frac{7}{2} - \frac{3}{2} \cdot 1 & - \frac{11}{2} - \frac{3}{2} \cdot - 1 & - \frac{111}{2} - \frac{3}{2} \cdot - 31 \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.8.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & 0 & 0 & 0 & 0 & - 1 & - 7 & 21 \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.9

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 5$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \\ 0 & 0 & 0 & 0 & 0 & - 1 & - 7 & 21 \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.10

Multiply each element of $R_{3}$ by $\frac{1}{3}$ to make the entry at $3, 5$ a $1$ .

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Step 2.10.1

Multiply each element of $R_{3}$ by $\frac{1}{3}$ to make the entry at $3, 5$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ \frac{0}{3} & \frac{0}{3} & \frac{0}{3} & \frac{0}{3} & \frac{3}{3} & \frac{2}{3} & \frac{- 4}{3} & \frac{- 9}{3} \\ 0 & 0 & 0 & 0 & 0 & - 1 & - 7 & 21 \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.10.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & 0 & 0 & 0 & 1 & \frac{2}{3} & - \frac{4}{3} & - 3 \\ 0 & 0 & 0 & 0 & 0 & - 1 & - 7 & 21 \\ 0 & 0 & 0 & 0 & 3 & 2 & - 4 & - 9 \end{array}]$

Step 2.11

Perform the row operation $R_{5} = R_{5} - 3 R_{3}$ to make the entry at $5, 5$ a $0$ .

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Step 2.11.1

Perform the row operation $R_{5} = R_{5} - 3 R_{3}$ to make the entry at $5, 5$ a $0$ .

Step 2.11.2

Simplify $R_{5}$ .

Step 2.12

Multiply each element of $R_{4}$ by $- 1$ to make the entry at $4, 6$ a $1$ .

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Step 2.12.1

Multiply each element of $R_{4}$ by $- 1$ to make the entry at $4, 6$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & 0 & 0 & 0 & 1 & \frac{2}{3} & - \frac{4}{3} & - 3 \\ - 0 & - 0 & - 0 & - 0 & - 0 & - - 1 & - - 7 & - 1 \cdot 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.12.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & 0 & 0 & 0 & 1 & \frac{2}{3} & - \frac{4}{3} & - 3 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.13

Perform the row operation $R_{3} = R_{3} - \frac{2}{3} R_{4}$ to make the entry at $3, 6$ a $0$ .

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Step 2.13.1

Perform the row operation $R_{3} = R_{3} - \frac{2}{3} R_{4}$ to make the entry at $3, 6$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 - \frac{2}{3} \cdot 0 & 0 - \frac{2}{3} \cdot 0 & 0 - \frac{2}{3} \cdot 0 & 0 - \frac{2}{3} \cdot 0 & 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & - \frac{4}{3} - \frac{2}{3} \cdot 7 & - 3 - \frac{2}{3} \cdot - 21 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.13.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 1 & - 1 & - 31 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.14

Perform the row operation $R_{2} = R_{2} - R_{4}$ to make the entry at $2, 6$ a $0$ .

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Step 2.14.1

Perform the row operation $R_{2} = R_{2} - R_{4}$ to make the entry at $2, 6$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 - 0 & 1 - 0 & - 5 - 0 & 4 - 0 & 0 - 0 & 1 - 1 & - 1 - 7 & - 31 + 21 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.14.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & \frac{1}{2} & \frac{5}{2} & \frac{21}{2} \\ 0 & 1 & - 5 & 4 & 0 & 0 & - 8 & - 10 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.15

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{4}$ to make the entry at $1, 6$ a $0$ .

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Step 2.15.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{4}$ to make the entry at $1, 6$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 0 & - \frac{1}{2} - \frac{1}{2} \cdot 0 & - 1 - \frac{1}{2} \cdot 0 & 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & \frac{5}{2} - \frac{1}{2} \cdot 7 & \frac{21}{2} - \frac{1}{2} \cdot - 21 \\ 0 & 1 & - 5 & 4 & 0 & 0 & - 8 & - 10 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.15.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 1 & 0 & - 1 & 21 \\ 0 & 1 & - 5 & 4 & 0 & 0 & - 8 & - 10 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.16

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 5$ a $0$ .

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Step 2.16.1

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 5$ a $0$ .

$[\begin{array}{c} 1 - 0 & \frac{1}{2} - 0 & - \frac{1}{2} - 0 & - 1 - 0 & 1 - 1 & 0 - 0 & - 1 + 6 & 21 - 11 \\ 0 & 1 & - 5 & 4 & 0 & 0 & - 8 & - 10 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.16.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & - 1 & 0 & 0 & 5 & 10 \\ 0 & 1 & - 5 & 4 & 0 & 0 & - 8 & - 10 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.17

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 2.17.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & - \frac{1}{2} - \frac{1}{2} \cdot - 5 & - 1 - \frac{1}{2} \cdot 4 & 0 - \frac{1}{2} \cdot 0 & 0 - \frac{1}{2} \cdot 0 & 5 - \frac{1}{2} \cdot - 8 & 10 - \frac{1}{2} \cdot - 10 \\ 0 & 1 & - 5 & 4 & 0 & 0 & - 8 & - 10 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.17.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 2 & - 3 & 0 & 0 & 9 & 15 \\ 0 & 1 & - 5 & 4 & 0 & 0 & - 8 & - 10 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 6 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & - 21 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 3

Use the result matrix to declare the final solutions to the system of equations.

$a + 2 d - 3 g + 9 k = 15$

$b - 5 d + 4 g - 8 k = - 10$

$h - 6 k = 11$

$j + 7 k = - 21$

Step 4

Move all terms not containing $a$ to the right side of the equation.

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Step 4.1

Subtract $2 d$ from both sides of the equation.

$a - 3 g + 9 k = 15 - 2 d$

$b - 5 d + 4 g - 8 k = - 10$

$h - 6 k = 11$

$j + 7 k = - 21$

Step 4.2

Add $3 g$ to both sides of the equation.

$a + 9 k = 15 - 2 d + 3 g$

$b - 5 d + 4 g - 8 k = - 10$

$h - 6 k = 11$

$j + 7 k = - 21$

Step 4.3

Subtract $9 k$ from both sides of the equation.

$a = 15 - 2 d + 3 g - 9 k$

$b - 5 d + 4 g - 8 k = - 10$

$h - 6 k = 11$

$j + 7 k = - 21$

$a = 15 - 2 d + 3 g - 9 k$

$b - 5 d + 4 g - 8 k = - 10$

$h - 6 k = 11$

$j + 7 k = - 21$

Step 5

Move all terms not containing $b$ to the right side of the equation.

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Step 5.1

Add $5 d$ to both sides of the equation.

$b + 4 g - 8 k = - 10 + 5 d$

$a = 15 - 2 d + 3 g - 9 k$

$h - 6 k = 11$

$j + 7 k = - 21$

Step 5.2

Subtract $4 g$ from both sides of the equation.

$b - 8 k = - 10 + 5 d - 4 g$

$a = 15 - 2 d + 3 g - 9 k$

$h - 6 k = 11$

$j + 7 k = - 21$

Step 5.3

Add $8 k$ to both sides of the equation.

$b = - 10 + 5 d - 4 g + 8 k$

$a = 15 - 2 d + 3 g - 9 k$

$h - 6 k = 11$

$j + 7 k = - 21$

$b = - 10 + 5 d - 4 g + 8 k$

$a = 15 - 2 d + 3 g - 9 k$

$h - 6 k = 11$

$j + 7 k = - 21$

Step 6

Add $6 k$ to both sides of the equation.

$h = 11 + 6 k$

$a = 15 - 2 d + 3 g - 9 k$

$b = - 10 + 5 d - 4 g + 8 k$

$j + 7 k = - 21$

Step 7

Subtract $7 k$ from both sides of the equation.

$j = - 21 - 7 k$

$a = 15 - 2 d + 3 g - 9 k$

$b = - 10 + 5 d - 4 g + 8 k$

$h = 11 + 6 k$

Step 8

The solution is the set of ordered pairs that makes the system true.

$(15 - 2 d + 3 g - 9 k, - 10 + 5 d - 4 g + 8 k, d, g, 11 + 6 k, - 21 - 7 k, k)$

Step 9

Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.

$X = [\begin{array}{c} a \\ b \\ d \\ g \\ h \\ j \\ k \end{array}] = [\begin{array}{c} 15 - 2 d + 3 g - 9 k \\ - 10 + 5 d - 4 g + 8 k \\ d \\ g \\ 11 + 6 k \\ - 21 - 7 k \\ k \end{array}]$