Конечная математика Примеры

$35 x + 70 y + 175 d + 70 w = 2240$ , $40 x + 40 y + 80 d + 70 w = 1120$ , $40 x + 30 y + 70 d + 10 w = 920$ , $w = 1$

Этап 1

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

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Этап 1.1

Перенесем $70 y$ .

$35 x + 175 d + 70 w + 70 y = 2240$

$40 x + 40 y + 80 d + 70 w = 1120$

$40 x + 30 y + 70 d + 10 w = 920$

$w = 1$

Этап 1.2

Перенесем $35 x$ .

$175 d + 70 w + 35 x + 70 y = 2240$

$40 x + 40 y + 80 d + 70 w = 1120$

$40 x + 30 y + 70 d + 10 w = 920$

$w = 1$

Этап 1.3

Перенесем $40 y$ .

$175 d + 70 w + 35 x + 70 y = 2240$

$40 x + 80 d + 70 w + 40 y = 1120$

$40 x + 30 y + 70 d + 10 w = 920$

$w = 1$

Этап 1.4

Перенесем $40 x$ .

$175 d + 70 w + 35 x + 70 y = 2240$

$80 d + 70 w + 40 x + 40 y = 1120$

$40 x + 30 y + 70 d + 10 w = 920$

$w = 1$

Этап 1.5

Перенесем $30 y$ .

$175 d + 70 w + 35 x + 70 y = 2240$

$80 d + 70 w + 40 x + 40 y = 1120$

$40 x + 70 d + 10 w + 30 y = 920$

$w = 1$

Этап 1.6

Перенесем $40 x$ .

$175 d + 70 w + 35 x + 70 y = 2240$

$80 d + 70 w + 40 x + 40 y = 1120$

$70 d + 10 w + 40 x + 30 y = 920$

$w = 1$

$175 d + 70 w + 35 x + 70 y = 2240$

$80 d + 70 w + 40 x + 40 y = 1120$

$70 d + 10 w + 40 x + 30 y = 920$

$w = 1$

Этап 2

Write the system as a matrix.

$[\begin{array}{c} 175 & 70 & 35 & 70 & 2240 \\ 80 & 70 & 40 & 40 & 1120 \\ 70 & 10 & 40 & 30 & 920 \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3

Приведем матрицу к стандартной форме по строкам.

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Этап 3.1

Multiply each element of $R_{1}$ by $\frac{1}{175}$ to make the entry at $1, 1$ a $1$ .

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Этап 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{175}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{175}{175} & \frac{70}{175} & \frac{35}{175} & \frac{70}{175} & \frac{2240}{175} \\ 80 & 70 & 40 & 40 & 1120 \\ 70 & 10 & 40 & 30 & 920 \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.1.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 80 & 70 & 40 & 40 & 1120 \\ 70 & 10 & 40 & 30 & 920 \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.2

Perform the row operation $R_{2} = R_{2} - 80 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Этап 3.2.1

Perform the row operation $R_{2} = R_{2} - 80 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 80 - 80 \cdot 1 & 70 - 80 (\frac{2}{5}) & 40 - 80 (\frac{1}{5}) & 40 - 80 (\frac{2}{5}) & 1120 - 80 (\frac{64}{5}) \\ 70 & 10 & 40 & 30 & 920 \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.2.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 38 & 24 & 8 & 96 \\ 70 & 10 & 40 & 30 & 920 \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.3

Perform the row operation $R_{3} = R_{3} - 70 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Этап 3.3.1

Perform the row operation $R_{3} = R_{3} - 70 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 38 & 24 & 8 & 96 \\ 70 - 70 \cdot 1 & 10 - 70 (\frac{2}{5}) & 40 - 70 (\frac{1}{5}) & 30 - 70 (\frac{2}{5}) & 920 - 70 (\frac{64}{5}) \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.3.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 38 & 24 & 8 & 96 \\ 0 & - 18 & 26 & 2 & 24 \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.4

Multiply each element of $R_{2}$ by $\frac{1}{38}$ to make the entry at $2, 2$ a $1$ .

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Этап 3.4.1

Multiply each element of $R_{2}$ by $\frac{1}{38}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ \frac{0}{38} & \frac{38}{38} & \frac{24}{38} & \frac{8}{38} & \frac{96}{38} \\ 0 & - 18 & 26 & 2 & 24 \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.4.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & \frac{4}{19} & \frac{48}{19} \\ 0 & - 18 & 26 & 2 & 24 \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.5

Perform the row operation $R_{3} = R_{3} + 18 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Этап 3.5.1

Perform the row operation $R_{3} = R_{3} + 18 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & \frac{4}{19} & \frac{48}{19} \\ 0 + 18 \cdot 0 & - 18 + 18 \cdot 1 & 26 + 18 (\frac{12}{19}) & 2 + 18 (\frac{4}{19}) & 24 + 18 (\frac{48}{19}) \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.5.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & \frac{4}{19} & \frac{48}{19} \\ 0 & 0 & \frac{710}{19} & \frac{110}{19} & \frac{1320}{19} \\ 0 & 1 & 0 & 0 & 1 \end{array}]$

Этап 3.6

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

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Этап 3.6.1

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

Этап 3.6.2

Упростим $R_{4}$ .

Этап 3.7

Multiply each element of $R_{3}$ by $\frac{19}{710}$ to make the entry at $3, 3$ a $1$ .

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Этап 3.7.1

Multiply each element of $R_{3}$ by $\frac{19}{710}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & \frac{4}{19} & \frac{48}{19} \\ \frac{19}{710} \cdot 0 & \frac{19}{710} \cdot 0 & \frac{19}{710} \cdot \frac{710}{19} & \frac{19}{710} \cdot \frac{110}{19} & \frac{19}{710} \cdot \frac{1320}{19} \\ 0 & 0 & - \frac{12}{19} & - \frac{4}{19} & - \frac{29}{19} \end{array}]$

Этап 3.7.2

Упростим $R_{3}$ .

Этап 3.8

Perform the row operation $R_{4} = R_{4} + \frac{12}{19} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Этап 3.8.1

Perform the row operation $R_{4} = R_{4} + \frac{12}{19} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & \frac{4}{19} & \frac{48}{19} \\ 0 & 0 & 1 & \frac{11}{71} & \frac{132}{71} \\ 0 + \frac{12}{19} \cdot 0 & 0 + \frac{12}{19} \cdot 0 & - \frac{12}{19} + \frac{12}{19} \cdot 1 & - \frac{4}{19} + \frac{12}{19} \cdot \frac{11}{71} & - \frac{29}{19} + \frac{12}{19} \cdot \frac{132}{71} \end{array}]$

Этап 3.8.2

Упростим $R_{4}$ .

Этап 3.9

Multiply each element of $R_{4}$ by $- \frac{71}{8}$ to make the entry at $4, 4$ a $1$ .

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Этап 3.9.1

Multiply each element of $R_{4}$ by $- \frac{71}{8}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & \frac{4}{19} & \frac{48}{19} \\ 0 & 0 & 1 & \frac{11}{71} & \frac{132}{71} \\ - \frac{71}{8} \cdot 0 & - \frac{71}{8} \cdot 0 & - \frac{71}{8} \cdot 0 & - \frac{71}{8} (- \frac{8}{71}) & - \frac{71}{8} (- \frac{25}{71}) \end{array}]$

Этап 3.9.2

Упростим $R_{4}$ .

Этап 3.10

Perform the row operation $R_{3} = R_{3} - \frac{11}{71} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Этап 3.10.1

Perform the row operation $R_{3} = R_{3} - \frac{11}{71} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & \frac{4}{19} & \frac{48}{19} \\ 0 - \frac{11}{71} \cdot 0 & 0 - \frac{11}{71} \cdot 0 & 1 - \frac{11}{71} \cdot 0 & \frac{11}{71} - \frac{11}{71} \cdot 1 & \frac{132}{71} - \frac{11}{71} \cdot \frac{25}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.10.2

Упростим $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & \frac{4}{19} & \frac{48}{19} \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.11

Perform the row operation $R_{2} = R_{2} - \frac{4}{19} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Этап 3.11.1

Perform the row operation $R_{2} = R_{2} - \frac{4}{19} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 - \frac{4}{19} \cdot 0 & 1 - \frac{4}{19} \cdot 0 & \frac{12}{19} - \frac{4}{19} \cdot 0 & \frac{4}{19} - \frac{4}{19} \cdot 1 & \frac{48}{19} - \frac{4}{19} \cdot \frac{25}{8} \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.11.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & \frac{2}{5} & \frac{64}{5} \\ 0 & 1 & \frac{12}{19} & 0 & \frac{71}{38} \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.12

Perform the row operation $R_{1} = R_{1} - \frac{2}{5} R_{4}$ to make the entry at $1, 4$ a $0$ .

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Этап 3.12.1

Perform the row operation $R_{1} = R_{1} - \frac{2}{5} R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{2}{5} \cdot 0 & \frac{2}{5} - \frac{2}{5} \cdot 0 & \frac{1}{5} - \frac{2}{5} \cdot 0 & \frac{2}{5} - \frac{2}{5} \cdot 1 & \frac{64}{5} - \frac{2}{5} \cdot \frac{25}{8} \\ 0 & 1 & \frac{12}{19} & 0 & \frac{71}{38} \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.12.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & 0 & \frac{231}{20} \\ 0 & 1 & \frac{12}{19} & 0 & \frac{71}{38} \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.13

Perform the row operation $R_{2} = R_{2} - \frac{12}{19} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Этап 3.13.1

Perform the row operation $R_{2} = R_{2} - \frac{12}{19} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & 0 & \frac{231}{20} \\ 0 - \frac{12}{19} \cdot 0 & 1 - \frac{12}{19} \cdot 0 & \frac{12}{19} - \frac{12}{19} \cdot 1 & 0 - \frac{12}{19} \cdot 0 & \frac{71}{38} - \frac{12}{19} \cdot \frac{11}{8} \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.13.2

Упростим $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{1}{5} & 0 & \frac{231}{20} \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.14

Perform the row operation $R_{1} = R_{1} - \frac{1}{5} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Этап 3.14.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{5} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{5} \cdot 0 & \frac{2}{5} - \frac{1}{5} \cdot 0 & \frac{1}{5} - \frac{1}{5} \cdot 1 & 0 - \frac{1}{5} \cdot 0 & \frac{231}{20} - \frac{1}{5} \cdot \frac{11}{8} \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.14.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 0 & 0 & \frac{451}{40} \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.15

Perform the row operation $R_{1} = R_{1} - \frac{2}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Этап 3.15.1

Perform the row operation $R_{1} = R_{1} - \frac{2}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{2}{5} \cdot 0 & \frac{2}{5} - \frac{2}{5} \cdot 1 & 0 - \frac{2}{5} \cdot 0 & 0 - \frac{2}{5} \cdot 0 & \frac{451}{40} - \frac{2}{5} \cdot 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 3.15.2

Упростим $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & \frac{87}{8} \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & \frac{11}{8} \\ 0 & 0 & 0 & 1 & \frac{25}{8} \end{array}]$

Этап 4

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

$d = \frac{87}{8}$

$w = 1$

$x = \frac{11}{8}$

$y = \frac{25}{8}$

Этап 5

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

$(\frac{87}{8}, 1, \frac{11}{8}, \frac{25}{8})$