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$x + y - z = 1$ , $x + 3 z = - 5$ , $3 y - z = 5$

Bước 1

Write the system as a matrix.

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

Bước 2

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 2.1

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

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Bước 2.1.1

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

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

Bước 2.1.2

Rút gọn $R_{2}$ .

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

Bước 2.2

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

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Bước 2.2.1

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

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

Bước 2.2.2

Rút gọn $R_{2}$ .

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

Bước 2.3

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

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Bước 2.3.1

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

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

Bước 2.3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 1 & 1 \\ 0 & 1 & - 4 & 6 \\ 0 & 0 & 11 & - 13 \end{array}]$

Bước 2.4

Multiply each element of $R_{3}$ by $\frac{1}{11}$ to make the entry at $3, 3$ a $1$ .

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Bước 2.4.1

Multiply each element of $R_{3}$ by $\frac{1}{11}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 1 & 1 \\ 0 & 1 & - 4 & 6 \\ \frac{0}{11} & \frac{0}{11} & \frac{11}{11} & \frac{- 13}{11} \end{array}]$

Bước 2.4.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 1 & 1 \\ 0 & 1 & - 4 & 6 \\ 0 & 0 & 1 & - \frac{13}{11} \end{array}]$

Bước 2.5

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

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Bước 2.5.1

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

$[\begin{array}{c} 1 & 1 & - 1 & 1 \\ 0 + 4 \cdot 0 & 1 + 4 \cdot 0 & - 4 + 4 \cdot 1 & 6 + 4 (- \frac{13}{11}) \\ 0 & 0 & 1 & - \frac{13}{11} \end{array}]$

Bước 2.5.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & \frac{14}{11} \\ 0 & 0 & 1 & - \frac{13}{11} \end{array}]$

Bước 2.6

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

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Bước 2.6.1

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

$[\begin{array}{c} 1 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 1 - \frac{13}{11} \\ 0 & 1 & 0 & \frac{14}{11} \\ 0 & 0 & 1 & - \frac{13}{11} \end{array}]$

Bước 2.6.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & - \frac{2}{11} \\ 0 & 1 & 0 & \frac{14}{11} \\ 0 & 0 & 1 & - \frac{13}{11} \end{array}]$

Bước 2.7

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

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Bước 2.7.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & - \frac{2}{11} - \frac{14}{11} \\ 0 & 1 & 0 & \frac{14}{11} \\ 0 & 0 & 1 & - \frac{13}{11} \end{array}]$

Bước 2.7.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{16}{11} \\ 0 & 1 & 0 & \frac{14}{11} \\ 0 & 0 & 1 & - \frac{13}{11} \end{array}]$

Bước 3

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

$x = - \frac{16}{11}$

$y = \frac{14}{11}$

$z = - \frac{13}{11}$

Bước 4

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

$(- \frac{16}{11}, \frac{14}{11}, - \frac{13}{11})$