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$8 x - 2 y + 9 z = - 4$ , $8 x - 2 y + 4 z = 4$ , $4 y - 5 z = - 9$

Bước 1

Write the system as a matrix.

$[\begin{array}{c} 8 & - 2 & 9 & - 4 \\ 8 & - 2 & 4 & 4 \\ 0 & 4 & - 5 & - 9 \end{array}]$

Bước 2

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 2.1

Multiply each element of $R_{1}$ by $\frac{1}{8}$ to make the entry at $1, 1$ a $1$ .

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Bước 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{8}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{8}{8} & \frac{- 2}{8} & \frac{9}{8} & \frac{- 4}{8} \\ 8 & - 2 & 4 & 4 \\ 0 & 4 & - 5 & - 9 \end{array}]$

Bước 2.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 8 & - 2 & 4 & 4 \\ 0 & 4 & - 5 & - 9 \end{array}]$

Bước 2.2

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Bước 2.2.1

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 8 - 8 \cdot 1 & - 2 - 8 (- \frac{1}{4}) & 4 - 8 (\frac{9}{8}) & 4 - 8 (- \frac{1}{2}) \\ 0 & 4 & - 5 & - 9 \end{array}]$

Bước 2.2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 0 & 0 & - 5 & 8 \\ 0 & 4 & - 5 & - 9 \end{array}]$

Bước 2.3

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 0 & 4 & - 5 & - 9 \\ 0 & 0 & - 5 & 8 \end{array}]$

Bước 2.4

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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Bước 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ \frac{0}{4} & \frac{4}{4} & \frac{- 5}{4} & \frac{- 9}{4} \\ 0 & 0 & - 5 & 8 \end{array}]$

Bước 2.4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 0 & 1 & - \frac{5}{4} & - \frac{9}{4} \\ 0 & 0 & - 5 & 8 \end{array}]$

Bước 2.5

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

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Bước 2.5.1

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

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 0 & 1 & - \frac{5}{4} & - \frac{9}{4} \\ - \frac{1}{5} \cdot 0 & - \frac{1}{5} \cdot 0 & - \frac{1}{5} \cdot - 5 & - \frac{1}{5} \cdot 8 \end{array}]$

Bước 2.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 0 & 1 & - \frac{5}{4} & - \frac{9}{4} \\ 0 & 0 & 1 & - \frac{8}{5} \end{array}]$

Bước 2.6

Perform the row operation $R_{2} = R_{2} + \frac{5}{4} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Bước 2.6.1

Perform the row operation $R_{2} = R_{2} + \frac{5}{4} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 0 + \frac{5}{4} \cdot 0 & 1 + \frac{5}{4} \cdot 0 & - \frac{5}{4} + \frac{5}{4} \cdot 1 & - \frac{9}{4} + \frac{5}{4} (- \frac{8}{5}) \\ 0 & 0 & 1 & - \frac{8}{5} \end{array}]$

Bước 2.6.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & \frac{9}{8} & - \frac{1}{2} \\ 0 & 1 & 0 & - \frac{17}{4} \\ 0 & 0 & 1 & - \frac{8}{5} \end{array}]$

Bước 2.7

Perform the row operation $R_{1} = R_{1} - \frac{9}{8} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Bước 2.7.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{8} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{8} \cdot 0 & - \frac{1}{4} - \frac{9}{8} \cdot 0 & \frac{9}{8} - \frac{9}{8} \cdot 1 & - \frac{1}{2} - \frac{9}{8} (- \frac{8}{5}) \\ 0 & 1 & 0 & - \frac{17}{4} \\ 0 & 0 & 1 & - \frac{8}{5} \end{array}]$

Bước 2.7.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & \frac{13}{10} \\ 0 & 1 & 0 & - \frac{17}{4} \\ 0 & 0 & 1 & - \frac{8}{5} \end{array}]$

Bước 2.8

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

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Bước 2.8.1

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

$[\begin{array}{c} 1 + \frac{1}{4} \cdot 0 & - \frac{1}{4} + \frac{1}{4} \cdot 1 & 0 + \frac{1}{4} \cdot 0 & \frac{13}{10} + \frac{1}{4} (- \frac{17}{4}) \\ 0 & 1 & 0 & - \frac{17}{4} \\ 0 & 0 & 1 & - \frac{8}{5} \end{array}]$

Bước 2.8.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{19}{80} \\ 0 & 1 & 0 & - \frac{17}{4} \\ 0 & 0 & 1 & - \frac{8}{5} \end{array}]$

Bước 3

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

$x = \frac{19}{80}$

$y = - \frac{17}{4}$

$z = - \frac{8}{5}$

Bước 4

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

$(\frac{19}{80}, - \frac{17}{4}, - \frac{8}{5})$