الرياضيات المتناهية الأمثلة

$- x + 2 y + 2 z = 15$ $x - 5 y + 4 z = 9$ $5 x - 5 y + 4 z = 5$

خطوة 1

Write the system as a matrix.

$[\begin{array}{c} - 1 & 2 & 2 & 15 \\ 1 & - 5 & 4 & 9 \\ 5 & - 5 & 4 & 5 \end{array}]$

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 2.1

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

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خطوة 2.1.1

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

$[\begin{array}{c} - - 1 & - 1 \cdot 2 & - 1 \cdot 2 & - 1 \cdot 15 \\ 1 & - 5 & 4 & 9 \\ 5 & - 5 & 4 & 5 \end{array}]$

خطوة 2.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 1 & - 5 & 4 & 9 \\ 5 & - 5 & 4 & 5 \end{array}]$

خطوة 2.2

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

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خطوة 2.2.1

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

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 1 - 1 & - 5 + 2 & 4 + 2 & 9 + 15 \\ 5 & - 5 & 4 & 5 \end{array}]$

خطوة 2.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & - 3 & 6 & 24 \\ 5 & - 5 & 4 & 5 \end{array}]$

خطوة 2.3

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

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خطوة 2.3.1

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

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & - 3 & 6 & 24 \\ 5 - 5 \cdot 1 & - 5 - 5 \cdot - 2 & 4 - 5 \cdot - 2 & 5 - 5 \cdot - 15 \end{array}]$

خطوة 2.3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & - 3 & 6 & 24 \\ 0 & 5 & 14 & 80 \end{array}]$

خطوة 2.4

Multiply each element of $R_{2}$ by $- \frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

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خطوة 2.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 6 & - \frac{1}{3} \cdot 24 \\ 0 & 5 & 14 & 80 \end{array}]$

خطوة 2.4.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & 1 & - 2 & - 8 \\ 0 & 5 & 14 & 80 \end{array}]$

خطوة 2.5

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

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خطوة 2.5.1

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

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & 1 & - 2 & - 8 \\ 0 - 5 \cdot 0 & 5 - 5 \cdot 1 & 14 - 5 \cdot - 2 & 80 - 5 \cdot - 8 \end{array}]$

خطوة 2.5.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & 1 & - 2 & - 8 \\ 0 & 0 & 24 & 120 \end{array}]$

خطوة 2.6

Multiply each element of $R_{3}$ by $\frac{1}{24}$ to make the entry at $3, 3$ a $1$ .

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خطوة 2.6.1

Multiply each element of $R_{3}$ by $\frac{1}{24}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & 1 & - 2 & - 8 \\ \frac{0}{24} & \frac{0}{24} & \frac{24}{24} & \frac{120}{24} \end{array}]$

خطوة 2.6.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & 1 & - 2 & - 8 \\ 0 & 0 & 1 & 5 \end{array}]$

خطوة 2.7

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 2.7.1

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

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 8 + 2 \cdot 5 \\ 0 & 0 & 1 & 5 \end{array}]$

خطوة 2.7.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 2 & - 15 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 5 \end{array}]$

خطوة 2.8

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

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خطوة 2.8.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 15 + 2 \cdot 5 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 5 \end{array}]$

خطوة 2.8.2

بسّط $R_{1}$ .

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

خطوة 2.9

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

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خطوة 2.9.1

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

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

خطوة 2.9.2

بسّط $R_{1}$ .

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

خطوة 3

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

$x = - 1$

$y = 2$

$z = 5$

خطوة 4

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

$(- 1, 2, 5)$