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$2 x + 3 y + z = 9$ , $4 x + y - 3 z = - 7$ , $6 x + 2 y - 4 z = - 8$

Bước 1

Write the system as a matrix.

$[\begin{array}{c} 2 & 3 & 1 & 9 \\ 4 & 1 & - 3 & - 7 \\ 6 & 2 & - 4 & - 8 \end{array}]$

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 $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Bước 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 4 & 1 & - 3 & - 7 \\ 6 & 2 & - 4 & - 8 \end{array}]$

Bước 2.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 4 & 1 & - 3 & - 7 \\ 6 & 2 & - 4 & - 8 \end{array}]$

Bước 2.2

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

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Bước 2.2.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 4 - 4 \cdot 1 & 1 - 4 (\frac{3}{2}) & - 3 - 4 (\frac{1}{2}) & - 7 - 4 (\frac{9}{2}) \\ 6 & 2 & - 4 & - 8 \end{array}]$

Bước 2.2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 0 & - 5 & - 5 & - 25 \\ 6 & 2 & - 4 & - 8 \end{array}]$

Bước 2.3

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

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Bước 2.3.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 0 & - 5 & - 5 & - 25 \\ 6 - 6 \cdot 1 & 2 - 6 (\frac{3}{2}) & - 4 - 6 (\frac{1}{2}) & - 8 - 6 (\frac{9}{2}) \end{array}]$

Bước 2.3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 0 & - 5 & - 5 & - 25 \\ 0 & - 7 & - 7 & - 35 \end{array}]$

Bước 2.4

Multiply each element of $R_{2}$ by $- \frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Bước 2.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ - \frac{1}{5} \cdot 0 & - \frac{1}{5} \cdot - 5 & - \frac{1}{5} \cdot - 5 & - \frac{1}{5} \cdot - 25 \\ 0 & - 7 & - 7 & - 35 \end{array}]$

Bước 2.4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 0 & 1 & 1 & 5 \\ 0 & - 7 & - 7 & - 35 \end{array}]$

Bước 2.5

Perform the row operation $R_{3} = R_{3} + 7 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 2.5.1

Perform the row operation $R_{3} = R_{3} + 7 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 0 & 1 & 1 & 5 \\ 0 + 7 \cdot 0 & - 7 + 7 \cdot 1 & - 7 + 7 \cdot 1 & - 35 + 7 \cdot 5 \end{array}]$

Bước 2.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & \frac{1}{2} & \frac{9}{2} \\ 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 2.6

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

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Bước 2.6.1

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

$[\begin{array}{c} 1 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & \frac{1}{2} - \frac{3}{2} \cdot 1 & \frac{9}{2} - \frac{3}{2} \cdot 5 \\ 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 2.6.2

Rút gọn $R_{1}$ .

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

Bước 3

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

$x - z = - 3$

$y + z = 5$

$0 = 0$

Bước 4

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

$(- 3 + z, 5 - z, z)$