الرياضيات المتناهية الأمثلة

$[\begin{array}{c} 5 & 1 & 1 & 0 & 0 & 0 & 30 \\ 1 & 3 & 0 & 1 & 0 & 0 & 50 \\ 1 & 1 & 0 & 0 & 1 & 0 & 40 \\ - 2 & - 3 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1

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

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خطوة 1.1

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

$[\begin{array}{c} \frac{5}{5} & \frac{1}{5} & \frac{1}{5} & \frac{0}{5} & \frac{0}{5} & \frac{0}{5} & \frac{30}{5} \\ 1 & 3 & 0 & 1 & 0 & 0 & 50 \\ 1 & 1 & 0 & 0 & 1 & 0 & 40 \\ - 2 & - 3 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 1 & 3 & 0 & 1 & 0 & 0 & 50 \\ 1 & 1 & 0 & 0 & 1 & 0 & 40 \\ - 2 & - 3 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 2

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

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خطوة 2.1

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

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 1 - 1 & 3 - \frac{1}{5} & 0 - \frac{1}{5} & 1 - 0 & 0 - 0 & 0 - 0 & 50 - 6 \\ 1 & 1 & 0 & 0 & 1 & 0 & 40 \\ - 2 & - 3 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & \frac{14}{5} & - \frac{1}{5} & 1 & 0 & 0 & 44 \\ 1 & 1 & 0 & 0 & 1 & 0 & 40 \\ - 2 & - 3 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 3

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

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خطوة 3.1

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

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & \frac{14}{5} & - \frac{1}{5} & 1 & 0 & 0 & 44 \\ 1 - 1 & 1 - \frac{1}{5} & 0 - \frac{1}{5} & 0 - 0 & 1 - 0 & 0 - 0 & 40 - 6 \\ - 2 & - 3 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & \frac{14}{5} & - \frac{1}{5} & 1 & 0 & 0 & 44 \\ 0 & \frac{4}{5} & - \frac{1}{5} & 0 & 1 & 0 & 34 \\ - 2 & - 3 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 4

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

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خطوة 4.1

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

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & \frac{14}{5} & - \frac{1}{5} & 1 & 0 & 0 & 44 \\ 0 & \frac{4}{5} & - \frac{1}{5} & 0 & 1 & 0 & 34 \\ - 2 + 2 \cdot 1 & - 3 + 2 (\frac{1}{5}) & 0 + 2 (\frac{1}{5}) & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & 0 + 2 \cdot 6 \end{array}]$

خطوة 4.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & \frac{14}{5} & - \frac{1}{5} & 1 & 0 & 0 & 44 \\ 0 & \frac{4}{5} & - \frac{1}{5} & 0 & 1 & 0 & 34 \\ 0 & - \frac{13}{5} & \frac{2}{5} & 0 & 0 & 1 & 12 \end{array}]$

خطوة 5

Multiply each element of $R_{2}$ by $\frac{5}{14}$ to make the entry at $2, 2$ a $1$ .

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خطوة 5.1

Multiply each element of $R_{2}$ by $\frac{5}{14}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ \frac{5}{14} \cdot 0 & \frac{5}{14} \cdot \frac{14}{5} & \frac{5}{14} (- \frac{1}{5}) & \frac{5}{14} \cdot 1 & \frac{5}{14} \cdot 0 & \frac{5}{14} \cdot 0 & \frac{5}{14} \cdot 44 \\ 0 & \frac{4}{5} & - \frac{1}{5} & 0 & 1 & 0 & 34 \\ 0 & - \frac{13}{5} & \frac{2}{5} & 0 & 0 & 1 & 12 \end{array}]$

خطوة 5.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 & \frac{4}{5} & - \frac{1}{5} & 0 & 1 & 0 & 34 \\ 0 & - \frac{13}{5} & \frac{2}{5} & 0 & 0 & 1 & 12 \end{array}]$

خطوة 6

Perform the row operation $R_{3} = R_{3} - \frac{4}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

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خطوة 6.1

Perform the row operation $R_{3} = R_{3} - \frac{4}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 - \frac{4}{5} \cdot 0 & \frac{4}{5} - \frac{4}{5} \cdot 1 & - \frac{1}{5} - \frac{4}{5} (- \frac{1}{14}) & 0 - \frac{4}{5} \cdot \frac{5}{14} & 1 - \frac{4}{5} \cdot 0 & 0 - \frac{4}{5} \cdot 0 & 34 - \frac{4}{5} \cdot \frac{110}{7} \\ 0 & - \frac{13}{5} & \frac{2}{5} & 0 & 0 & 1 & 12 \end{array}]$

خطوة 6.2

بسّط $R_{3}$ .

خطوة 7

Perform the row operation $R_{4} = R_{4} + \frac{13}{5} R_{2}$ to make the entry at $4, 2$ a $0$ .

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خطوة 7.1

Perform the row operation $R_{4} = R_{4} + \frac{13}{5} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 & 0 & - \frac{1}{7} & - \frac{2}{7} & 1 & 0 & \frac{150}{7} \\ 0 + \frac{13}{5} \cdot 0 & - \frac{13}{5} + \frac{13}{5} \cdot 1 & \frac{2}{5} + \frac{13}{5} (- \frac{1}{14}) & 0 + \frac{13}{5} \cdot \frac{5}{14} & 0 + \frac{13}{5} \cdot 0 & 1 + \frac{13}{5} \cdot 0 & 12 + \frac{13}{5} \cdot \frac{110}{7} \end{array}]$

خطوة 7.2

بسّط $R_{4}$ .

خطوة 8

Multiply each element of $R_{3}$ by $- 7$ to make the entry at $3, 3$ a $1$ .

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خطوة 8.1

Multiply each element of $R_{3}$ by $- 7$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ - 7 \cdot 0 & - 7 \cdot 0 & - 7 (- \frac{1}{7}) & - 7 (- \frac{2}{7}) & - 7 \cdot 1 & - 7 \cdot 0 & - 7 (\frac{150}{7}) \\ 0 & 0 & \frac{3}{14} & \frac{13}{14} & 0 & 1 & \frac{370}{7} \end{array}]$

خطوة 8.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 & 0 & 1 & 2 & - 7 & 0 & - 150 \\ 0 & 0 & \frac{3}{14} & \frac{13}{14} & 0 & 1 & \frac{370}{7} \end{array}]$

خطوة 9

Perform the row operation $R_{4} = R_{4} - \frac{3}{14} R_{3}$ to make the entry at $4, 3$ a $0$ .

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خطوة 9.1

Perform the row operation $R_{4} = R_{4} - \frac{3}{14} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 & 0 & 1 & 2 & - 7 & 0 & - 150 \\ 0 - \frac{3}{14} \cdot 0 & 0 - \frac{3}{14} \cdot 0 & \frac{3}{14} - \frac{3}{14} \cdot 1 & \frac{13}{14} - \frac{3}{14} \cdot 2 & 0 - \frac{3}{14} \cdot - 7 & 1 - \frac{3}{14} \cdot 0 & \frac{370}{7} - \frac{3}{14} \cdot - 150 \end{array}]$

خطوة 9.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 & 0 & 1 & 2 & - 7 & 0 & - 150 \\ 0 & 0 & 0 & \frac{1}{2} & \frac{3}{2} & 1 & 85 \end{array}]$

خطوة 10

Multiply each element of $R_{4}$ by $2$ to make the entry at $4, 4$ a $1$ .

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خطوة 10.1

Multiply each element of $R_{4}$ by $2$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 & 0 & 1 & 2 & - 7 & 0 & - 150 \\ 2 \cdot 0 & 2 \cdot 0 & 2 \cdot 0 & 2 (\frac{1}{2}) & 2 (\frac{3}{2}) & 2 \cdot 1 & 2 \cdot 85 \end{array}]$

خطوة 10.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 & 0 & 1 & 2 & - 7 & 0 & - 150 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 11

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

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خطوة 11.1

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

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 7 - 2 \cdot 3 & 0 - 2 \cdot 2 & - 150 - 2 \cdot 170 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 11.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & \frac{5}{14} & 0 & 0 & \frac{110}{7} \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 12

Perform the row operation $R_{2} = R_{2} - \frac{5}{14} R_{4}$ to make the entry at $2, 4$ a $0$ .

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خطوة 12.1

Perform the row operation $R_{2} = R_{2} - \frac{5}{14} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 - \frac{5}{14} \cdot 0 & 1 - \frac{5}{14} \cdot 0 & - \frac{1}{14} - \frac{5}{14} \cdot 0 & \frac{5}{14} - \frac{5}{14} \cdot 1 & 0 - \frac{5}{14} \cdot 3 & 0 - \frac{5}{14} \cdot 2 & \frac{110}{7} - \frac{5}{14} \cdot 170 \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 12.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & - \frac{1}{14} & 0 & - \frac{15}{14} & - \frac{5}{7} & - 45 \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 13

Perform the row operation $R_{2} = R_{2} + \frac{1}{14} R_{3}$ to make the entry at $2, 3$ a $0$ .

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خطوة 13.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{14} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 + \frac{1}{14} \cdot 0 & 1 + \frac{1}{14} \cdot 0 & - \frac{1}{14} + \frac{1}{14} \cdot 1 & 0 + \frac{1}{14} \cdot 0 & - \frac{15}{14} + \frac{1}{14} \cdot - 13 & - \frac{5}{7} + \frac{1}{14} \cdot - 4 & - 45 + \frac{1}{14} \cdot - 490 \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 13.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 6 \\ 0 & 1 & 0 & 0 & - 2 & - 1 & - 80 \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 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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خطوة 14.1

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

$[\begin{array}{c} 1 - \frac{1}{5} \cdot 0 & \frac{1}{5} - \frac{1}{5} \cdot 0 & \frac{1}{5} - \frac{1}{5} \cdot 1 & 0 - \frac{1}{5} \cdot 0 & 0 - \frac{1}{5} \cdot - 13 & 0 - \frac{1}{5} \cdot - 4 & 6 - \frac{1}{5} \cdot - 490 \\ 0 & 1 & 0 & 0 & - 2 & - 1 & - 80 \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 14.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{5} & 0 & 0 & \frac{13}{5} & \frac{4}{5} & 104 \\ 0 & 1 & 0 & 0 & - 2 & - 1 & - 80 \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 15

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

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خطوة 15.1

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

$[\begin{array}{c} 1 - \frac{1}{5} \cdot 0 & \frac{1}{5} - \frac{1}{5} \cdot 1 & 0 - \frac{1}{5} \cdot 0 & 0 - \frac{1}{5} \cdot 0 & \frac{13}{5} - \frac{1}{5} \cdot - 2 & \frac{4}{5} - \frac{1}{5} \cdot - 1 & 104 - \frac{1}{5} \cdot - 80 \\ 0 & 1 & 0 & 0 & - 2 & - 1 & - 80 \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$

خطوة 15.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 3 & 1 & 120 \\ 0 & 1 & 0 & 0 & - 2 & - 1 & - 80 \\ 0 & 0 & 1 & 0 & - 13 & - 4 & - 490 \\ 0 & 0 & 0 & 1 & 3 & 2 & 170 \end{array}]$