Finite Math Examples

$x + y + z - 2 v + 3 w + u = 1$ , $- 2 x + 3 y - 2 z + v + 2 w + 3 u = 3$ , $2 x + 2 y - 4 z + 2 v + w + 3 u = 2$ , $- 2 x + y + z - 2 v - 2 w - 2 u = 1$ , $- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$ , $- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1

Move variables to the left and constant terms to the right.

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Step 1.1

Move $z$ .

$x + y - 2 v + 3 w + u + z = 1$

$- 2 x + 3 y - 2 z + v + 2 w + 3 u = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.2

Move $y$ .

$x - 2 v + 3 w + u + y + z = 1$

$- 2 x + 3 y - 2 z + v + 2 w + 3 u = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.3

Move $x$ .

$- 2 v + 3 w + u + x + y + z = 1$

$- 2 x + 3 y - 2 z + v + 2 w + 3 u = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.4

Move $3 w$ .

$- 2 v + u + 3 w + x + y + z = 1$

$- 2 x + 3 y - 2 z + v + 2 w + 3 u = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.5

Reorder $- 2 v$ and $u$ .

$u - 2 v + 3 w + x + y + z = 1$

$- 2 x + 3 y - 2 z + v + 2 w + 3 u = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.6

Move $- 2 z$ .

$u - 2 v + 3 w + x + y + z = 1$

$- 2 x + 3 y + v + 2 w + 3 u - 2 z = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.7

Move $3 y$ .

$u - 2 v + 3 w + x + y + z = 1$

$- 2 x + v + 2 w + 3 u + 3 y - 2 z = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.8

Move $- 2 x$ .

$u - 2 v + 3 w + x + y + z = 1$

$v + 2 w + 3 u - 2 x + 3 y - 2 z = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.9

Move $2 w$ .

$u - 2 v + 3 w + x + y + z = 1$

$v + 3 u + 2 w - 2 x + 3 y - 2 z = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.10

Reorder $v$ and $3 u$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$2 x + 2 y - 4 z + 2 v + w + 3 u = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.11

Move $- 4 z$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$2 x + 2 y + 2 v + w + 3 u - 4 z = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.12

Move $2 y$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$2 x + 2 v + w + 3 u + 2 y - 4 z = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.13

Move $2 x$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$2 v + w + 3 u + 2 x + 2 y - 4 z = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.14

Move $w$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$2 v + 3 u + w + 2 x + 2 y - 4 z = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.15

Reorder $2 v$ and $3 u$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 x + y + z - 2 v - 2 w - 2 u = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.16

Move $z$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 x + y - 2 v - 2 w - 2 u + z = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.17

Move $y$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 x - 2 v - 2 w - 2 u + y + z = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.18

Move $- 2 x$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 v - 2 w - 2 u - 2 x + y + z = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.19

Move $- 2 w$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 v - 2 u - 2 w - 2 x + y + z = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.20

Reorder $- 2 v$ and $- 2 u$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- 3 x + 2 y - 3 z + 2 v + 2 w - u = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.21

Move $- 3 z$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- 3 x + 2 y + 2 v + 2 w - u - 3 z = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.22

Move $2 y$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- 3 x + 2 v + 2 w - u + 2 y - 3 z = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.23

Move $- 3 x$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$2 v + 2 w - u - 3 x + 2 y - 3 z = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.24

Move $2 w$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$2 v - u + 2 w - 3 x + 2 y - 3 z = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.25

Reorder $2 v$ and $- u$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- u + 2 v + 2 w - 3 x + 2 y - 3 z = - 4$

$- x + y + 2 z + v - 2 w - 3 u = 3$

Step 1.26

Move $2 z$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- u + 2 v + 2 w - 3 x + 2 y - 3 z = - 4$

$- x + y + v - 2 w - 3 u + 2 z = 3$

Step 1.27

Move $y$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- u + 2 v + 2 w - 3 x + 2 y - 3 z = - 4$

$- x + v - 2 w - 3 u + y + 2 z = 3$

Step 1.28

Move $- x$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- u + 2 v + 2 w - 3 x + 2 y - 3 z = - 4$

$v - 2 w - 3 u - x + y + 2 z = 3$

Step 1.29

Move $- 2 w$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- u + 2 v + 2 w - 3 x + 2 y - 3 z = - 4$

$v - 3 u - 2 w - x + y + 2 z = 3$

Step 1.30

Reorder $v$ and $- 3 u$ .

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- u + 2 v + 2 w - 3 x + 2 y - 3 z = - 4$

$- 3 u + v - 2 w - x + y + 2 z = 3$

$u - 2 v + 3 w + x + y + z = 1$

$3 u + v + 2 w - 2 x + 3 y - 2 z = 3$

$3 u + 2 v + w + 2 x + 2 y - 4 z = 2$

$- 2 u - 2 v - 2 w - 2 x + y + z = 1$

$- u + 2 v + 2 w - 3 x + 2 y - 3 z = - 4$

$- 3 u + v - 2 w - x + y + 2 z = 3$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 3 & 1 & 2 & - 2 & 3 & - 2 & 3 \\ 3 & 2 & 1 & 2 & 2 & - 4 & 2 \\ - 2 & - 2 & - 2 & - 2 & 1 & 1 & 1 \\ - 1 & 2 & 2 & - 3 & 2 & - 3 & - 4 \\ - 3 & 1 & - 2 & - 1 & 1 & 2 & 3 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Perform the row operation $R_{2} = R_{2} - 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.1.1

Perform the row operation $R_{2} = R_{2} - 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 3 - 3 \cdot 1 & 1 - 3 \cdot - 2 & 2 - 3 \cdot 3 & - 2 - 3 \cdot 1 & 3 - 3 \cdot 1 & - 2 - 3 \cdot 1 & 3 - 3 \cdot 1 \\ 3 & 2 & 1 & 2 & 2 & - 4 & 2 \\ - 2 & - 2 & - 2 & - 2 & 1 & 1 & 1 \\ - 1 & 2 & 2 & - 3 & 2 & - 3 & - 4 \\ - 3 & 1 & - 2 & - 1 & 1 & 2 & 3 \end{array}]$

Step 3.1.2

Simplify $R_{2}$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 7 & - 7 & - 5 & 0 & - 5 & 0 \\ 0 & 8 & - 8 & - 1 & - 1 & - 7 & - 1 \\ - 2 & - 2 & - 2 & - 2 & 1 & 1 & 1 \\ - 1 & 2 & 2 & - 3 & 2 & - 3 & - 4 \\ - 3 & 1 & - 2 & - 1 & 1 & 2 & 3 \end{array}]$

Step 3.3

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

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Step 3.3.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 7 & - 7 & - 5 & 0 & - 5 & 0 \\ 0 & 8 & - 8 & - 1 & - 1 & - 7 & - 1 \\ - 2 + 2 \cdot 1 & - 2 + 2 \cdot - 2 & - 2 + 2 \cdot 3 & - 2 + 2 \cdot 1 & 1 + 2 \cdot 1 & 1 + 2 \cdot 1 & 1 + 2 \cdot 1 \\ - 1 & 2 & 2 & - 3 & 2 & - 3 & - 4 \\ - 3 & 1 & - 2 & - 1 & 1 & 2 & 3 \end{array}]$

Step 3.3.2

Simplify $R_{4}$ .

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

Step 3.4

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

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Step 3.4.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 7 & - 7 & - 5 & 0 & - 5 & 0 \\ 0 & 8 & - 8 & - 1 & - 1 & - 7 & - 1 \\ 0 & - 6 & 4 & 0 & 3 & 3 & 3 \\ - 1 + 1 \cdot 1 & 2 - 2 & 2 + 1 \cdot 3 & - 3 + 1 \cdot 1 & 2 + 1 \cdot 1 & - 3 + 1 \cdot 1 & - 4 + 1 \cdot 1 \\ - 3 & 1 & - 2 & - 1 & 1 & 2 & 3 \end{array}]$

Step 3.4.2

Simplify $R_{5}$ .

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

Step 3.5

Perform the row operation $R_{6} = R_{6} + 3 R_{1}$ to make the entry at $6, 1$ a $0$ .

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Step 3.5.1

Perform the row operation $R_{6} = R_{6} + 3 R_{1}$ to make the entry at $6, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 7 & - 7 & - 5 & 0 & - 5 & 0 \\ 0 & 8 & - 8 & - 1 & - 1 & - 7 & - 1 \\ 0 & - 6 & 4 & 0 & 3 & 3 & 3 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ - 3 + 3 \cdot 1 & 1 + 3 \cdot - 2 & - 2 + 3 \cdot 3 & - 1 + 3 \cdot 1 & 1 + 3 \cdot 1 & 2 + 3 \cdot 1 & 3 + 3 \cdot 1 \end{array}]$

Step 3.5.2

Simplify $R_{6}$ .

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

Step 3.6

Multiply each element of $R_{2}$ by $\frac{1}{7}$ to make the entry at $2, 2$ a $1$ .

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Step 3.6.1

Multiply each element of $R_{2}$ by $\frac{1}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ \frac{0}{7} & \frac{7}{7} & \frac{- 7}{7} & \frac{- 5}{7} & \frac{0}{7} & \frac{- 5}{7} & \frac{0}{7} \\ 0 & 8 & - 8 & - 1 & - 1 & - 7 & - 1 \\ 0 & - 6 & 4 & 0 & 3 & 3 & 3 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ 0 & - 5 & 7 & 2 & 4 & 5 & 6 \end{array}]$

Step 3.6.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 8 & - 8 & - 1 & - 1 & - 7 & - 1 \\ 0 & - 6 & 4 & 0 & 3 & 3 & 3 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ 0 & - 5 & 7 & 2 & 4 & 5 & 6 \end{array}]$

Step 3.7

Perform the row operation $R_{3} = R_{3} - 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 3.7.1

Perform the row operation $R_{3} = R_{3} - 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 - 8 \cdot 0 & 8 - 8 \cdot 1 & - 8 - 8 \cdot - 1 & - 1 - 8 (- \frac{5}{7}) & - 1 - 8 \cdot 0 & - 7 - 8 (- \frac{5}{7}) & - 1 - 8 \cdot 0 \\ 0 & - 6 & 4 & 0 & 3 & 3 & 3 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ 0 & - 5 & 7 & 2 & 4 & 5 & 6 \end{array}]$

Step 3.7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 & - 6 & 4 & 0 & 3 & 3 & 3 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ 0 & - 5 & 7 & 2 & 4 & 5 & 6 \end{array}]$

Step 3.8

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

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Step 3.8.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 & 4 + 6 \cdot - 1 & 0 + 6 (- \frac{5}{7}) & 3 + 6 \cdot 0 & 3 + 6 (- \frac{5}{7}) & 3 + 6 \cdot 0 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ 0 & - 5 & 7 & 2 & 4 & 5 & 6 \end{array}]$

Step 3.8.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 & 0 & - 2 & - \frac{30}{7} & 3 & - \frac{9}{7} & 3 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ 0 & - 5 & 7 & 2 & 4 & 5 & 6 \end{array}]$

Step 3.9

Perform the row operation $R_{6} = R_{6} + 5 R_{2}$ to make the entry at $6, 2$ a $0$ .

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Step 3.9.1

Perform the row operation $R_{6} = R_{6} + 5 R_{2}$ to make the entry at $6, 2$ a $0$ .

Step 3.9.2

Simplify $R_{6}$ .

Step 3.10

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & - 2 & - \frac{30}{7} & 3 & - \frac{9}{7} & 3 \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ 0 & 0 & 2 & - \frac{11}{7} & 4 & \frac{10}{7} & 6 \end{array}]$

Step 3.11

Multiply each element of $R_{3}$ by $- \frac{1}{2}$ to make the entry at $3, 3$ a $1$ .

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Step 3.11.1

Multiply each element of $R_{3}$ by $- \frac{1}{2}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} (- \frac{30}{7}) & - \frac{1}{2} \cdot 3 & - \frac{1}{2} (- \frac{9}{7}) & - \frac{1}{2} \cdot 3 \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 & 0 & 5 & - 2 & 3 & - 2 & - 3 \\ 0 & 0 & 2 & - \frac{11}{7} & 4 & \frac{10}{7} & 6 \end{array}]$

Step 3.11.2

Simplify $R_{3}$ .

Step 3.12

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

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Step 3.12.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 - 5 \cdot 0 & 0 - 5 \cdot 0 & 5 - 5 \cdot 1 & - 2 - 5 (\frac{15}{7}) & 3 - 5 (- \frac{3}{2}) & - 2 - 5 (\frac{9}{14}) & - 3 - 5 (- \frac{3}{2}) \\ 0 & 0 & 2 & - \frac{11}{7} & 4 & \frac{10}{7} & 6 \end{array}]$

Step 3.12.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 & 0 & 0 & - \frac{89}{7} & \frac{21}{2} & - \frac{73}{14} & \frac{9}{2} \\ 0 & 0 & 2 & - \frac{11}{7} & 4 & \frac{10}{7} & 6 \end{array}]$

Step 3.13

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

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Step 3.13.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 & 0 & 0 & - \frac{89}{7} & \frac{21}{2} & - \frac{73}{14} & \frac{9}{2} \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - \frac{11}{7} - 2 (\frac{15}{7}) & 4 - 2 (- \frac{3}{2}) & \frac{10}{7} - 2 (\frac{9}{14}) & 6 - 2 (- \frac{3}{2}) \end{array}]$

Step 3.13.2

Simplify $R_{6}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & \frac{33}{7} & - 1 & - \frac{9}{7} & - 1 \\ 0 & 0 & 0 & - \frac{89}{7} & \frac{21}{2} & - \frac{73}{14} & \frac{9}{2} \\ 0 & 0 & 0 & - \frac{41}{7} & 7 & \frac{1}{7} & 9 \end{array}]$

Step 3.14

Multiply each element of $R_{4}$ by $\frac{7}{33}$ to make the entry at $4, 4$ a $1$ .

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Step 3.14.1

Multiply each element of $R_{4}$ by $\frac{7}{33}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ \frac{7}{33} \cdot 0 & \frac{7}{33} \cdot 0 & \frac{7}{33} \cdot 0 & \frac{7}{33} \cdot \frac{33}{7} & \frac{7}{33} \cdot - 1 & \frac{7}{33} (- \frac{9}{7}) & \frac{7}{33} \cdot - 1 \\ 0 & 0 & 0 & - \frac{89}{7} & \frac{21}{2} & - \frac{73}{14} & \frac{9}{2} \\ 0 & 0 & 0 & - \frac{41}{7} & 7 & \frac{1}{7} & 9 \end{array}]$

Step 3.14.2

Simplify $R_{4}$ .

Step 3.15

Perform the row operation $R_{5} = R_{5} + \frac{89}{7} R_{4}$ to make the entry at $5, 4$ a $0$ .

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Step 3.15.1

Perform the row operation $R_{5} = R_{5} + \frac{89}{7} R_{4}$ to make the entry at $5, 4$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & - \frac{3}{11} & - \frac{7}{33} \\ 0 + \frac{89}{7} \cdot 0 & 0 + \frac{89}{7} \cdot 0 & 0 + \frac{89}{7} \cdot 0 & - \frac{89}{7} + \frac{89}{7} \cdot 1 & \frac{21}{2} + \frac{89}{7} (- \frac{7}{33}) & - \frac{73}{14} + \frac{89}{7} (- \frac{3}{11}) & \frac{9}{2} + \frac{89}{7} (- \frac{7}{33}) \\ 0 & 0 & 0 & - \frac{41}{7} & 7 & \frac{1}{7} & 9 \end{array}]$

Step 3.15.2

Simplify $R_{5}$ .

Step 3.16

Perform the row operation $R_{6} = R_{6} + \frac{41}{7} R_{4}$ to make the entry at $6, 4$ a $0$ .

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Step 3.16.1

Perform the row operation $R_{6} = R_{6} + \frac{41}{7} R_{4}$ to make the entry at $6, 4$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & - \frac{3}{11} & - \frac{7}{33} \\ 0 & 0 & 0 & 0 & \frac{515}{66} & - \frac{191}{22} & \frac{119}{66} \\ 0 + \frac{41}{7} \cdot 0 & 0 + \frac{41}{7} \cdot 0 & 0 + \frac{41}{7} \cdot 0 & - \frac{41}{7} + \frac{41}{7} \cdot 1 & 7 + \frac{41}{7} (- \frac{7}{33}) & \frac{1}{7} + \frac{41}{7} (- \frac{3}{11}) & 9 + \frac{41}{7} (- \frac{7}{33}) \end{array}]$

Step 3.16.2

Simplify $R_{6}$ .

Step 3.17

Multiply each element of $R_{5}$ by $\frac{66}{515}$ to make the entry at $5, 5$ a $1$ .

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Step 3.17.1

Multiply each element of $R_{5}$ by $\frac{66}{515}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & - \frac{3}{11} & - \frac{7}{33} \\ \frac{66}{515} \cdot 0 & \frac{66}{515} \cdot 0 & \frac{66}{515} \cdot 0 & \frac{66}{515} \cdot 0 & \frac{66}{515} \cdot \frac{515}{66} & \frac{66}{515} (- \frac{191}{22}) & \frac{66}{515} \cdot \frac{119}{66} \\ 0 & 0 & 0 & 0 & \frac{190}{33} & - \frac{16}{11} & \frac{256}{33} \end{array}]$

Step 3.17.2

Simplify $R_{5}$ .

Step 3.18

Perform the row operation $R_{6} = R_{6} - \frac{190}{33} R_{5}$ to make the entry at $6, 5$ a $0$ .

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Step 3.18.1

Perform the row operation $R_{6} = R_{6} - \frac{190}{33} R_{5}$ to make the entry at $6, 5$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & - \frac{3}{11} & - \frac{7}{33} \\ 0 & 0 & 0 & 0 & 1 & - \frac{573}{515} & \frac{119}{515} \\ 0 - \frac{190}{33} \cdot 0 & 0 - \frac{190}{33} \cdot 0 & 0 - \frac{190}{33} \cdot 0 & 0 - \frac{190}{33} \cdot 0 & \frac{190}{33} - \frac{190}{33} \cdot 1 & - \frac{16}{11} - \frac{190}{33} (- \frac{573}{515}) & \frac{256}{33} - \frac{190}{33} \cdot \frac{119}{515} \end{array}]$

Step 3.18.2

Simplify $R_{6}$ .

Step 3.19

Multiply each element of $R_{6}$ by $\frac{103}{510}$ to make the entry at $6, 6$ a $1$ .

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Step 3.19.1

Multiply each element of $R_{6}$ by $\frac{103}{510}$ to make the entry at $6, 6$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & - \frac{3}{11} & - \frac{7}{33} \\ 0 & 0 & 0 & 0 & 1 & - \frac{573}{515} & \frac{119}{515} \\ \frac{103}{510} \cdot 0 & \frac{103}{510} \cdot 0 & \frac{103}{510} \cdot 0 & \frac{103}{510} \cdot 0 & \frac{103}{510} \cdot 0 & \frac{103}{510} \cdot \frac{510}{103} & \frac{103}{510} \cdot \frac{662}{103} \end{array}]$

Step 3.19.2

Simplify $R_{6}$ .

Step 3.20

Perform the row operation $R_{5} = R_{5} + \frac{573}{515} R_{6}$ to make the entry at $5, 6$ a $0$ .

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Step 3.20.1

Perform the row operation $R_{5} = R_{5} + \frac{573}{515} R_{6}$ to make the entry at $5, 6$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & - \frac{3}{11} & - \frac{7}{33} \\ 0 + \frac{573}{515} \cdot 0 & 0 + \frac{573}{515} \cdot 0 & 0 + \frac{573}{515} \cdot 0 & 0 + \frac{573}{515} \cdot 0 & 1 + \frac{573}{515} \cdot 0 & - \frac{573}{515} + \frac{573}{515} \cdot 1 & \frac{119}{515} + \frac{573}{515} \cdot \frac{331}{255} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.20.2

Simplify $R_{5}$ .

Step 3.21

Perform the row operation $R_{4} = R_{4} + \frac{3}{11} R_{6}$ to make the entry at $4, 6$ a $0$ .

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Step 3.21.1

Perform the row operation $R_{4} = R_{4} + \frac{3}{11} R_{6}$ to make the entry at $4, 6$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & \frac{9}{14} & - \frac{3}{2} \\ 0 + \frac{3}{11} \cdot 0 & 0 + \frac{3}{11} \cdot 0 & 0 + \frac{3}{11} \cdot 0 & 1 + \frac{3}{11} \cdot 0 & - \frac{7}{33} + \frac{3}{11} \cdot 0 & - \frac{3}{11} + \frac{3}{11} \cdot 1 & - \frac{7}{33} + \frac{3}{11} \cdot \frac{331}{255} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.21.2

Simplify $R_{4}$ .

Step 3.22

Perform the row operation $R_{3} = R_{3} - \frac{9}{14} R_{6}$ to make the entry at $3, 6$ a $0$ .

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Step 3.22.1

Perform the row operation $R_{3} = R_{3} - \frac{9}{14} R_{6}$ to make the entry at $3, 6$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 - \frac{9}{14} \cdot 0 & 0 - \frac{9}{14} \cdot 0 & 1 - \frac{9}{14} \cdot 0 & \frac{15}{7} - \frac{9}{14} \cdot 0 & - \frac{3}{2} - \frac{9}{14} \cdot 0 & \frac{9}{14} - \frac{9}{14} \cdot 1 & - \frac{3}{2} - \frac{9}{14} \cdot \frac{331}{255} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & 0 & \frac{398}{2805} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.22.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & - \frac{5}{7} & 0 \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & 0 & - \frac{1389}{595} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & 0 & \frac{398}{2805} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.23

Perform the row operation $R_{2} = R_{2} + \frac{5}{7} R_{6}$ to make the entry at $2, 6$ a $0$ .

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Step 3.23.1

Perform the row operation $R_{2} = R_{2} + \frac{5}{7} R_{6}$ to make the entry at $2, 6$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 + \frac{5}{7} \cdot 0 & 1 + \frac{5}{7} \cdot 0 & - 1 + \frac{5}{7} \cdot 0 & - \frac{5}{7} + \frac{5}{7} \cdot 0 & 0 + \frac{5}{7} \cdot 0 & - \frac{5}{7} + \frac{5}{7} \cdot 1 & 0 + \frac{5}{7} \cdot \frac{331}{255} \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & 0 & - \frac{1389}{595} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & 0 & \frac{398}{2805} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.23.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 1 & 1 \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & 0 & - \frac{1389}{595} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & 0 & \frac{398}{2805} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.24

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

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Step 3.24.1

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

$[\begin{array}{c} 1 - 0 & - 2 - 0 & 3 - 0 & 1 - 0 & 1 - 0 & 1 - 1 & 1 - \frac{331}{255} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & 0 & - \frac{1389}{595} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & 0 & \frac{398}{2805} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.24.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 0 & - \frac{76}{255} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & 0 & - \frac{1389}{595} \\ 0 & 0 & 0 & 1 & - \frac{7}{33} & 0 & \frac{398}{2805} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.25

Perform the row operation $R_{4} = R_{4} + \frac{7}{33} R_{5}$ to make the entry at $4, 5$ a $0$ .

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Step 3.25.1

Perform the row operation $R_{4} = R_{4} + \frac{7}{33} R_{5}$ to make the entry at $4, 5$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 0 & - \frac{76}{255} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & 0 & - \frac{1389}{595} \\ 0 + \frac{7}{33} \cdot 0 & 0 + \frac{7}{33} \cdot 0 & 0 + \frac{7}{33} \cdot 0 & 1 + \frac{7}{33} \cdot 0 & - \frac{7}{33} + \frac{7}{33} \cdot 1 & 0 + \frac{7}{33} \cdot 0 & \frac{398}{2805} + \frac{7}{33} \cdot \frac{712}{425} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.25.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 0 & - \frac{76}{255} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & \frac{15}{7} & - \frac{3}{2} & 0 & - \frac{1389}{595} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.26

Perform the row operation $R_{3} = R_{3} + \frac{3}{2} R_{5}$ to make the entry at $3, 5$ a $0$ .

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Step 3.26.1

Perform the row operation $R_{3} = R_{3} + \frac{3}{2} R_{5}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 0 & - \frac{76}{255} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 + \frac{3}{2} \cdot 0 & 0 + \frac{3}{2} \cdot 0 & 1 + \frac{3}{2} \cdot 0 & \frac{15}{7} + \frac{3}{2} \cdot 0 & - \frac{3}{2} + \frac{3}{2} \cdot 1 & 0 + \frac{3}{2} \cdot 0 & - \frac{1389}{595} + \frac{3}{2} \cdot \frac{712}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.26.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 1 & 0 & - \frac{76}{255} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & \frac{15}{7} & 0 & 0 & \frac{531}{2975} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.27

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

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Step 3.27.1

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

$[\begin{array}{c} 1 - 0 & - 2 - 0 & 3 - 0 & 1 - 0 & 1 - 1 & 0 - 0 & - \frac{76}{255} - \frac{712}{425} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & \frac{15}{7} & 0 & 0 & \frac{531}{2975} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.27.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 0 & 0 & - \frac{148}{75} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & \frac{15}{7} & 0 & 0 & \frac{531}{2975} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.28

Perform the row operation $R_{3} = R_{3} - \frac{15}{7} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 3.28.1

Perform the row operation $R_{3} = R_{3} - \frac{15}{7} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 0 & 0 & - \frac{148}{75} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 - \frac{15}{7} \cdot 0 & 0 - \frac{15}{7} \cdot 0 & 1 - \frac{15}{7} \cdot 0 & \frac{15}{7} - \frac{15}{7} \cdot 1 & 0 - \frac{15}{7} \cdot 0 & 0 - \frac{15}{7} \cdot 0 & \frac{531}{2975} - \frac{15}{7} \cdot \frac{634}{1275} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.28.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 0 & 0 & - \frac{148}{75} \\ 0 & 1 & - 1 & - \frac{5}{7} & 0 & 0 & \frac{331}{357} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.29

Perform the row operation $R_{2} = R_{2} + \frac{5}{7} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 3.29.1

Perform the row operation $R_{2} = R_{2} + \frac{5}{7} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 0 & 0 & - \frac{148}{75} \\ 0 + \frac{5}{7} \cdot 0 & 1 + \frac{5}{7} \cdot 0 & - 1 + \frac{5}{7} \cdot 0 & - \frac{5}{7} + \frac{5}{7} \cdot 1 & 0 + \frac{5}{7} \cdot 0 & 0 + \frac{5}{7} \cdot 0 & \frac{331}{357} + \frac{5}{7} \cdot \frac{634}{1275} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.29.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 1 & 0 & 0 & - \frac{148}{75} \\ 0 & 1 & - 1 & 0 & 0 & 0 & \frac{109}{85} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.30

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

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Step 3.30.1

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

$[\begin{array}{c} 1 - 0 & - 2 - 0 & 3 - 0 & 1 - 1 & 0 - 0 & 0 - 0 & - \frac{148}{75} - \frac{634}{1275} \\ 0 & 1 & - 1 & 0 & 0 & 0 & \frac{109}{85} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.30.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 0 & 0 & 0 & - \frac{42}{17} \\ 0 & 1 & - 1 & 0 & 0 & 0 & \frac{109}{85} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.31

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

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Step 3.31.1

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

$[\begin{array}{c} 1 & - 2 & 3 & 0 & 0 & 0 & - \frac{42}{17} \\ 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 + 0 & 0 + 0 & \frac{109}{85} - \frac{377}{425} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.31.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 3 & 0 & 0 & 0 & - \frac{42}{17} \\ 0 & 1 & 0 & 0 & 0 & 0 & \frac{168}{425} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.32

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

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Step 3.32.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & - 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & 0 - 3 \cdot 0 & 0 - 3 \cdot 0 & 0 - 3 \cdot 0 & - \frac{42}{17} - 3 (- \frac{377}{425}) \\ 0 & 1 & 0 & 0 & 0 & 0 & \frac{168}{425} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.32.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 & 0 & 0 & \frac{81}{425} \\ 0 & 1 & 0 & 0 & 0 & 0 & \frac{168}{425} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.33

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

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Step 3.33.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & \frac{81}{425} + 2 (\frac{168}{425}) \\ 0 & 1 & 0 & 0 & 0 & 0 & \frac{168}{425} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 3.33.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & \frac{417}{425} \\ 0 & 1 & 0 & 0 & 0 & 0 & \frac{168}{425} \\ 0 & 0 & 1 & 0 & 0 & 0 & - \frac{377}{425} \\ 0 & 0 & 0 & 1 & 0 & 0 & \frac{634}{1275} \\ 0 & 0 & 0 & 0 & 1 & 0 & \frac{712}{425} \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{331}{255} \end{array}]$

Step 4

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

$u = \frac{417}{425}$

$v = \frac{168}{425}$

$w = - \frac{377}{425}$

$x = \frac{634}{1275}$

$y = \frac{712}{425}$

$z = \frac{331}{255}$

Step 5

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

$(\frac{417}{425}, \frac{168}{425}, - \frac{377}{425}, \frac{634}{1275}, \frac{712}{425}, \frac{331}{255})$