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$400 x + 1200 y + 800 z = 8000$ , $110 x + 570 y + 340 z = 2920$ , $90 x + 30 y + 60 z = 1080$

Bước 1

Write the system as a matrix.

$[\begin{array}{c} 400 & 1200 & 800 & 8000 \\ 110 & 570 & 340 & 2920 \\ 90 & 30 & 60 & 1080 \end{array}]$

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 $\frac{1}{400}$ to make the entry at $1, 1$ a $1$ .

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Bước 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{400}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{400}{400} & \frac{1200}{400} & \frac{800}{400} & \frac{8000}{400} \\ 110 & 570 & 340 & 2920 \\ 90 & 30 & 60 & 1080 \end{array}]$

Bước 2.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ 110 & 570 & 340 & 2920 \\ 90 & 30 & 60 & 1080 \end{array}]$

Bước 2.2

Perform the row operation $R_{2} = R_{2} - 110 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Bước 2.2.1

Perform the row operation $R_{2} = R_{2} - 110 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ 110 - 110 \cdot 1 & 570 - 110 \cdot 3 & 340 - 110 \cdot 2 & 2920 - 110 \cdot 20 \\ 90 & 30 & 60 & 1080 \end{array}]$

Bước 2.2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ 0 & 240 & 120 & 720 \\ 90 & 30 & 60 & 1080 \end{array}]$

Bước 2.3

Perform the row operation $R_{3} = R_{3} - 90 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 2.3.1

Perform the row operation $R_{3} = R_{3} - 90 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ 0 & 240 & 120 & 720 \\ 90 - 90 \cdot 1 & 30 - 90 \cdot 3 & 60 - 90 \cdot 2 & 1080 - 90 \cdot 20 \end{array}]$

Bước 2.3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ 0 & 240 & 120 & 720 \\ 0 & - 240 & - 120 & - 720 \end{array}]$

Bước 2.4

Multiply each element of $R_{2}$ by $\frac{1}{240}$ to make the entry at $2, 2$ a $1$ .

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Bước 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{240}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ \frac{0}{240} & \frac{240}{240} & \frac{120}{240} & \frac{720}{240} \\ 0 & - 240 & - 120 & - 720 \end{array}]$

Bước 2.4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ 0 & 1 & \frac{1}{2} & 3 \\ 0 & - 240 & - 120 & - 720 \end{array}]$

Bước 2.5

Perform the row operation $R_{3} = R_{3} + 240 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 2.5.1

Perform the row operation $R_{3} = R_{3} + 240 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ 0 & 1 & \frac{1}{2} & 3 \\ 0 + 240 \cdot 0 & - 240 + 240 \cdot 1 & - 120 + 240 (\frac{1}{2}) & - 720 + 240 \cdot 3 \end{array}]$

Bước 2.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 3 & 2 & 20 \\ 0 & 1 & \frac{1}{2} & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 2.6

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

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Bước 2.6.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & 2 - 3 (\frac{1}{2}) & 20 - 3 \cdot 3 \\ 0 & 1 & \frac{1}{2} & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 2.6.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{2} & 11 \\ 0 & 1 & \frac{1}{2} & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Bước 3

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

$x + \frac{1}{2} z = 11$

$y + \frac{1}{2} z = 3$

$0 = 0$

Bước 4

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

$(11 - \frac{z}{2}, 3 - \frac{z}{2}, z)$