الرياضيات المتناهية الأمثلة

$- 2 x + 5 y + z = - 3$ , $5 x + 14 y - z = - 11$ , $7 x + 9 y - 2 z = - 5$

خطوة 1

Write the system as a matrix.

$[\begin{array}{c} - 2 & 5 & 1 & - 3 \\ 5 & 14 & - 1 & - 11 \\ 7 & 9 & - 2 & - 5 \end{array}]$

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 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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خطوة 2.1.1

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

$[\begin{array}{c} - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 5 & - \frac{1}{2} \cdot 1 & - \frac{1}{2} \cdot - 3 \\ 5 & 14 & - 1 & - 11 \\ 7 & 9 & - 2 & - 5 \end{array}]$

خطوة 2.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 5 & 14 & - 1 & - 11 \\ 7 & 9 & - 2 & - 5 \end{array}]$

خطوة 2.2

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

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خطوة 2.2.1

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

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 5 - 5 \cdot 1 & 14 - 5 (- \frac{5}{2}) & - 1 - 5 (- \frac{1}{2}) & - 11 - 5 (\frac{3}{2}) \\ 7 & 9 & - 2 & - 5 \end{array}]$

خطوة 2.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & \frac{53}{2} & \frac{3}{2} & - \frac{37}{2} \\ 7 & 9 & - 2 & - 5 \end{array}]$

خطوة 2.3

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

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خطوة 2.3.1

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & \frac{53}{2} & \frac{3}{2} & - \frac{37}{2} \\ 7 - 7 \cdot 1 & 9 - 7 (- \frac{5}{2}) & - 2 - 7 (- \frac{1}{2}) & - 5 - 7 (\frac{3}{2}) \end{array}]$

خطوة 2.3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & \frac{53}{2} & \frac{3}{2} & - \frac{37}{2} \\ 0 & \frac{53}{2} & \frac{3}{2} & - \frac{31}{2} \end{array}]$

خطوة 2.4

Multiply each element of $R_{2}$ by $\frac{2}{53}$ to make the entry at $2, 2$ a $1$ .

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خطوة 2.4.1

Multiply each element of $R_{2}$ by $\frac{2}{53}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ \frac{2}{53} \cdot 0 & \frac{2}{53} \cdot \frac{53}{2} & \frac{2}{53} \cdot \frac{3}{2} & \frac{2}{53} (- \frac{37}{2}) \\ 0 & \frac{53}{2} & \frac{3}{2} & - \frac{31}{2} \end{array}]$

خطوة 2.4.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & 1 & \frac{3}{53} & - \frac{37}{53} \\ 0 & \frac{53}{2} & \frac{3}{2} & - \frac{31}{2} \end{array}]$

خطوة 2.5

Perform the row operation $R_{3} = R_{3} - \frac{53}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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خطوة 2.5.1

Perform the row operation $R_{3} = R_{3} - \frac{53}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & 1 & \frac{3}{53} & - \frac{37}{53} \\ 0 - \frac{53}{2} \cdot 0 & \frac{53}{2} - \frac{53}{2} \cdot 1 & \frac{3}{2} - \frac{53}{2} \cdot \frac{3}{53} & - \frac{31}{2} - \frac{53}{2} (- \frac{37}{53}) \end{array}]$

خطوة 2.5.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & 1 & \frac{3}{53} & - \frac{37}{53} \\ 0 & 0 & 0 & 3 \end{array}]$

خطوة 2.6

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

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خطوة 2.6.1

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

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & 1 & \frac{3}{53} & - \frac{37}{53} \\ \frac{0}{3} & \frac{0}{3} & \frac{0}{3} & \frac{3}{3} \end{array}]$

خطوة 2.6.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & 1 & \frac{3}{53} & - \frac{37}{53} \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.7

Perform the row operation $R_{2} = R_{2} + \frac{37}{53} R_{3}$ to make the entry at $2, 4$ a $0$ .

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خطوة 2.7.1

Perform the row operation $R_{2} = R_{2} + \frac{37}{53} R_{3}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 + \frac{37}{53} \cdot 0 & 1 + \frac{37}{53} \cdot 0 & \frac{3}{53} + \frac{37}{53} \cdot 0 & - \frac{37}{53} + \frac{37}{53} \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.7.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & \frac{3}{2} \\ 0 & 1 & \frac{3}{53} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.8

Perform the row operation $R_{1} = R_{1} - \frac{3}{2} R_{3}$ to make the entry at $1, 4$ a $0$ .

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خطوة 2.8.1

Perform the row operation $R_{1} = R_{1} - \frac{3}{2} R_{3}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{3}{2} \cdot 0 & - \frac{5}{2} - \frac{3}{2} \cdot 0 & - \frac{1}{2} - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 \\ 0 & 1 & \frac{3}{53} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.8.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & - \frac{1}{2} & 0 \\ 0 & 1 & \frac{3}{53} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.9

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

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خطوة 2.9.1

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

$[\begin{array}{c} 1 + \frac{5}{2} \cdot 0 & - \frac{5}{2} + \frac{5}{2} \cdot 1 & - \frac{1}{2} + \frac{5}{2} \cdot \frac{3}{53} & 0 + \frac{5}{2} \cdot 0 \\ 0 & 1 & \frac{3}{53} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.9.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{19}{53} & 0 \\ 0 & 1 & \frac{3}{53} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 3

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

$x - \frac{19}{53} z = 0$

$y + \frac{3}{53} z = 0$

$0 = 1$

خطوة 4

The system is inconsistent so there is no solution.

$لا يوجد حل$