الرياضيات المتناهية الأمثلة

$x + y + z = 1230$ , $120 x + 150 y + 180 z = 32400$ , $180 x + 80 y + 80 z = 34200$ , $5 x + 6 y + 4 z = 800$

خطوة 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 120 & 150 & 180 & 32400 \\ 180 & 80 & 80 & 34200 \\ 5 & 6 & 4 & 800 \end{array}]$

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 2.1

Perform the row operation $R_{2} = R_{2} - 120 R_{1}$ to make the entry at $2, 1$ a $0$ .

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خطوة 2.1.1

Perform the row operation $R_{2} = R_{2} - 120 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 120 - 120 \cdot 1 & 150 - 120 \cdot 1 & 180 - 120 \cdot 1 & 32400 - 120 \cdot 1230 \\ 180 & 80 & 80 & 34200 \\ 5 & 6 & 4 & 800 \end{array}]$

خطوة 2.1.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 30 & 60 & - 115200 \\ 180 & 80 & 80 & 34200 \\ 5 & 6 & 4 & 800 \end{array}]$

خطوة 2.2

Perform the row operation $R_{3} = R_{3} - 180 R_{1}$ to make the entry at $3, 1$ a $0$ .

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خطوة 2.2.1

Perform the row operation $R_{3} = R_{3} - 180 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 30 & 60 & - 115200 \\ 180 - 180 \cdot 1 & 80 - 180 \cdot 1 & 80 - 180 \cdot 1 & 34200 - 180 \cdot 1230 \\ 5 & 6 & 4 & 800 \end{array}]$

خطوة 2.2.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 30 & 60 & - 115200 \\ 0 & - 100 & - 100 & - 187200 \\ 5 & 6 & 4 & 800 \end{array}]$

خطوة 2.3

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

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خطوة 2.3.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 30 & 60 & - 115200 \\ 0 & - 100 & - 100 & - 187200 \\ 5 - 5 \cdot 1 & 6 - 5 \cdot 1 & 4 - 5 \cdot 1 & 800 - 5 \cdot 1230 \end{array}]$

خطوة 2.3.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 30 & 60 & - 115200 \\ 0 & - 100 & - 100 & - 187200 \\ 0 & 1 & - 1 & - 5350 \end{array}]$

خطوة 2.4

Multiply each element of $R_{2}$ by $\frac{1}{30}$ to make the entry at $2, 2$ a $1$ .

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خطوة 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{30}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ \frac{0}{30} & \frac{30}{30} & \frac{60}{30} & \frac{- 115200}{30} \\ 0 & - 100 & - 100 & - 187200 \\ 0 & 1 & - 1 & - 5350 \end{array}]$

خطوة 2.4.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & - 100 & - 100 & - 187200 \\ 0 & 1 & - 1 & - 5350 \end{array}]$

خطوة 2.5

Perform the row operation $R_{3} = R_{3} + 100 R_{2}$ to make the entry at $3, 2$ a $0$ .

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خطوة 2.5.1

Perform the row operation $R_{3} = R_{3} + 100 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 + 100 \cdot 0 & - 100 + 100 \cdot 1 & - 100 + 100 \cdot 2 & - 187200 + 100 \cdot - 3840 \\ 0 & 1 & - 1 & - 5350 \end{array}]$

خطوة 2.5.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 100 & - 571200 \\ 0 & 1 & - 1 & - 5350 \end{array}]$

خطوة 2.6

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

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خطوة 2.6.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 100 & - 571200 \\ 0 - 0 & 1 - 1 & - 1 - 2 & - 5350 + 3840 \end{array}]$

خطوة 2.6.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 100 & - 571200 \\ 0 & 0 & - 3 & - 1510 \end{array}]$

خطوة 2.7

Multiply each element of $R_{3}$ by $\frac{1}{100}$ to make the entry at $3, 3$ a $1$ .

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خطوة 2.7.1

Multiply each element of $R_{3}$ by $\frac{1}{100}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ \frac{0}{100} & \frac{0}{100} & \frac{100}{100} & \frac{- 571200}{100} \\ 0 & 0 & - 3 & - 1510 \end{array}]$

خطوة 2.7.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 1 & - 5712 \\ 0 & 0 & - 3 & - 1510 \end{array}]$

خطوة 2.8

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

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خطوة 2.8.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 1 & - 5712 \\ 0 + 3 \cdot 0 & 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 1510 + 3 \cdot - 5712 \end{array}]$

خطوة 2.8.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 1 & - 5712 \\ 0 & 0 & 0 & - 18646 \end{array}]$

خطوة 2.9

Multiply each element of $R_{4}$ by $- \frac{1}{18646}$ to make the entry at $4, 4$ a $1$ .

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خطوة 2.9.1

Multiply each element of $R_{4}$ by $- \frac{1}{18646}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 1 & - 5712 \\ - \frac{1}{18646} \cdot 0 & - \frac{1}{18646} \cdot 0 & - \frac{1}{18646} \cdot 0 & - \frac{1}{18646} \cdot - 18646 \end{array}]$

خطوة 2.9.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 1 & - 5712 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.10

Perform the row operation $R_{3} = R_{3} + 5712 R_{4}$ to make the entry at $3, 4$ a $0$ .

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خطوة 2.10.1

Perform the row operation $R_{3} = R_{3} + 5712 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 + 5712 \cdot 0 & 0 + 5712 \cdot 0 & 1 + 5712 \cdot 0 & - 5712 + 5712 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.10.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & - 3840 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.11

Perform the row operation $R_{2} = R_{2} + 3840 R_{4}$ to make the entry at $2, 4$ a $0$ .

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خطوة 2.11.1

Perform the row operation $R_{2} = R_{2} + 3840 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 + 3840 \cdot 0 & 1 + 3840 \cdot 0 & 2 + 3840 \cdot 0 & - 3840 + 3840 \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.11.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1230 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.12

Perform the row operation $R_{1} = R_{1} - 1230 R_{4}$ to make the entry at $1, 4$ a $0$ .

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خطوة 2.12.1

Perform the row operation $R_{1} = R_{1} - 1230 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 1230 \cdot 0 & 1 - 1230 \cdot 0 & 1 - 1230 \cdot 0 & 1230 - 1230 \cdot 1 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.12.2

بسّط $R_{1}$ .

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

خطوة 2.13

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

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خطوة 2.13.1

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

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

خطوة 2.13.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.14

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

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خطوة 2.14.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 0 - 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.14.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.15

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

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خطوة 2.15.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 0 - 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 2.15.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

خطوة 3

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

$x = 0$

$y = 0$

$z = 0$

$0 = 1$

خطوة 4

The system is inconsistent so there is no solution.

$لا يوجد حل$