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$x + y + z + w = 250000$ , $0.025 x + 0.035 y + 0.04 z + 0.06 w = 10000$ , $z - 0.5 w = 0$ , $x + y = 150000$

Bước 1

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

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Bước 1.1

Di chuyển $z$ .

$x + y + w + z = 250000$

$0.025 x + 0.035 y + 0.04 z + 0.06 w = 10000$

$z - 0.5 w = 0$

$x + y = 150000$

Bước 1.2

Di chuyển $y$ .

$x + w + y + z = 250000$

$0.025 x + 0.035 y + 0.04 z + 0.06 w = 10000$

$z - 0.5 w = 0$

$x + y = 150000$

Bước 1.3

Sắp xếp lại $x$ và $w$ .

$w + x + y + z = 250000$

$0.025 x + 0.035 y + 0.04 z + 0.06 w = 10000$

$z - 0.5 w = 0$

$x + y = 150000$

Bước 1.4

Di chuyển $0.04 z$ .

$w + x + y + z = 250000$

$0.025 x + 0.035 y + 0.06 w + 0.04 z = 10000$

$z - 0.5 w = 0$

$x + y = 150000$

Bước 1.5

Di chuyển $0.035 y$ .

$w + x + y + z = 250000$

$0.025 x + 0.06 w + 0.035 y + 0.04 z = 10000$

$z - 0.5 w = 0$

$x + y = 150000$

Bước 1.6

Sắp xếp lại $0.025 x$ và $0.06 w$ .

$w + x + y + z = 250000$

$0.06 w + 0.025 x + 0.035 y + 0.04 z = 10000$

$z - 0.5 w = 0$

$x + y = 150000$

Bước 1.7

Sắp xếp lại $z$ và $- 0.5 w$ .

$w + x + y + z = 250000$

$0.06 w + 0.025 x + 0.035 y + 0.04 z = 10000$

$- 0.5 w + z = 0$

$x + y = 150000$

$w + x + y + z = 250000$

$0.06 w + 0.025 x + 0.035 y + 0.04 z = 10000$

$- 0.5 w + z = 0$

$x + y = 150000$

Bước 2

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0.06 & 0.025 & 0.035 & 0.04 & 10000 \\ - 0.5 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 3.1

Perform the row operation $R_{2} = R_{2} - 0.06 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Bước 3.1.1

Perform the row operation $R_{2} = R_{2} - 0.06 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0.06 - 0.06 \cdot 1 & 0.025 - 0.06 \cdot 1 & 0.035 - 0.06 \cdot 1 & 0.04 - 0.06 \cdot 1 & 10000 - 0.06 \cdot 250000 \\ - 0.5 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3.1.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & - 0.035 & - 0.025 & - 0.02 & - 5000 \\ - 0.5 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3.2

Perform the row operation $R_{3} = R_{3} + 0.5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 3.2.1

Perform the row operation $R_{3} = R_{3} + 0.5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & - 0.035 & - 0.025 & - 0.02 & - 5000 \\ - 0.5 + 0.5 \cdot 1 & 0 + 0.5 \cdot 1 & 0 + 0.5 \cdot 1 & 1 + 0.5 \cdot 1 & 0 + 0.5 \cdot 250000 \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3.2.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & - 0.035 & - 0.025 & - 0.02 & - 5000 \\ 0 & 0.5 & 0.5 & 1.5 & 125000 \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3.3

Multiply each element of $R_{2}$ by $\frac{1}{- 0.035}$ to make the entry at $2, 2$ a $1$ .

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Bước 3.3.1

Multiply each element of $R_{2}$ by $\frac{1}{- 0.035}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ \frac{0}{- 0.035} & \frac{- 0.035}{- 0.035} & \frac{- 0.025}{- 0.035} & \frac{- 0.02}{- 0.035} & \frac{- 5000}{- 0.035} \\ 0 & 0.5 & 0.5 & 1.5 & 125000 \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3.3.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 & 0.5 & 0.5 & 1.5 & 125000 \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3.4

Perform the row operation $R_{3} = R_{3} - 0.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 3.4.1

Perform the row operation $R_{3} = R_{3} - 0.5 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 - 0.5 \cdot 0 & 0.5 - 0.5 \cdot 1 & 0.5 - 0.5 \cdot 0.\overline{714285} & 1.5 - 0.5 \cdot 0.\overline{571428} & 125000 - 0.5 \cdot 142857.\overline{142857} \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3.4.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 & 0 & 0.\overline{142857} & 1.2\overline{142857} & 53571.\overline{428571} \\ 0 & 1 & 1 & 0 & 150000 \end{array}]$

Bước 3.5

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

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Bước 3.5.1

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

Bước 3.5.2

Rút gọn $R_{4}$ .

Bước 3.6

Multiply each element of $R_{3}$ by $\frac{1}{0.\overline{142857}}$ to make the entry at $3, 3$ a $1$ .

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Bước 3.6.1

Multiply each element of $R_{3}$ by $\frac{1}{0.\overline{142857}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ \frac{0}{0.\overline{142857}} & \frac{0}{0.\overline{142857}} & \frac{0.\overline{142857}}{0.\overline{142857}} & \frac{1.2\overline{142857}}{0.\overline{142857}} & \frac{53571.\overline{428571}}{0.\overline{142857}} \\ 0 & 0 & 0.\overline{285714} & - 0.\overline{571428} & 7142.\overline{857142} \end{array}]$

Bước 3.6.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 & 0 & 1 & 8.5 & 375000 \\ 0 & 0 & 0.\overline{285714} & - 0.\overline{571428} & 7142.\overline{857142} \end{array}]$

Bước 3.7

Perform the row operation $R_{4} = R_{4} - 0.\overline{285714} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Bước 3.7.1

Perform the row operation $R_{4} = R_{4} - 0.\overline{285714} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 & 0 & 1 & 8.5 & 375000 \\ 0 - 0.\overline{285714} \cdot 0 & 0 - 0.\overline{285714} \cdot 0 & 0.\overline{285714} - 0.\overline{285714} \cdot 1 & - 0.\overline{571428} - 0.\overline{285714} \cdot 8.5 & 7142.\overline{857142} - 0.\overline{285714} \cdot 375000 \end{array}]$

Bước 3.7.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 & 0 & 1 & 8.5 & 375000 \\ 0 & 0 & 0 & - 3 & - 100000 \end{array}]$

Bước 3.8

Multiply each element of $R_{4}$ by $\frac{1}{- 3}$ to make the entry at $4, 4$ a $1$ .

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Bước 3.8.1

Multiply each element of $R_{4}$ by $\frac{1}{- 3}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 & 0 & 1 & 8.5 & 375000 \\ \frac{0}{- 3} & \frac{0}{- 3} & \frac{0}{- 3} & \frac{- 3}{- 3} & \frac{- 100000}{- 3} \end{array}]$

Bước 3.8.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 & 0 & 1 & 8.5 & 375000 \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.9

Perform the row operation $R_{3} = R_{3} - 8.5 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Bước 3.9.1

Perform the row operation $R_{3} = R_{3} - 8.5 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 - 8.5 \cdot 0 & 0 - 8.5 \cdot 0 & 1 - 8.5 \cdot 0 & 8.5 - 8.5 \cdot 1 & 375000 - 8.5 \cdot 33333.\overline{3} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.9.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0.\overline{571428} & 142857.\overline{142857} \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.10

Perform the row operation $R_{2} = R_{2} - 0.\overline{571428} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Bước 3.10.1

Perform the row operation $R_{2} = R_{2} - 0.\overline{571428} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 - 0.\overline{571428} \cdot 0 & 1 - 0.\overline{571428} \cdot 0 & 0.\overline{714285} - 0.\overline{571428} \cdot 0 & 0.\overline{571428} - 0.\overline{571428} \cdot 1 & 142857.\overline{142857} - 0.\overline{571428} \cdot 33333.\overline{3} \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.10.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 250000 \\ 0 & 1 & 0.\overline{714285} & 0 & 123809.\overline{523809} \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.11

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

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Bước 3.11.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 0 & 1 - 1 & 250000 - 33333.\overline{3} \\ 0 & 1 & 0.\overline{714285} & 0 & 123809.\overline{523809} \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.11.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 216666.\overline{6} \\ 0 & 1 & 0.\overline{714285} & 0 & 123809.\overline{523809} \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.12

Perform the row operation $R_{2} = R_{2} - 0.\overline{714285} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Bước 3.12.1

Perform the row operation $R_{2} = R_{2} - 0.\overline{714285} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 216666.\overline{6} \\ 0 - 0.\overline{714285} \cdot 0 & 1 - 0.\overline{714285} \cdot 0 & 0.\overline{714285} - 0.\overline{714285} \cdot 1 & 0 - 0.\overline{714285} \cdot 0 & 123809.\overline{523809} - 0.\overline{714285} \cdot 91666.\overline{6} \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.12.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 216666.\overline{6} \\ 0 & 1 & 0 & 0 & 58333.33333333 \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.13

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

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Bước 3.13.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 0 + 0 & 216666.\overline{6} - 91666.\overline{6} \\ 0 & 1 & 0 & 0 & 58333.33333333 \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.13.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 0 & 125000 \\ 0 & 1 & 0 & 0 & 58333.33333333 \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.14

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

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Bước 3.14.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 + 0 & 0 + 0 & 125000 - 58333.33333333 \\ 0 & 1 & 0 & 0 & 58333.33333333 \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 3.14.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 66666.\overline{6} \\ 0 & 1 & 0 & 0 & 58333.33333333 \\ 0 & 0 & 1 & 0 & 91666.\overline{6} \\ 0 & 0 & 0 & 1 & 33333.\overline{3} \end{array}]$

Bước 4

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

$w = 66666.\overline{6}$

$x = 58333.33333333$

$y = 91666.\overline{6}$

$z = 33333.\overline{3}$

Bước 5

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

$(66666.\overline{6}, 58333.33333333, 91666.\overline{6}, 33333.\overline{3})$