الجبر الخطي الأمثلة

$a + 2 b + g = 7$ , $a + b + d - g = 3$ , $3 a + b + 5 d - 7 g = 1$

خطوة 1

Write the system as a matrix.

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

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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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 $0$ .

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خطوة 2.1.1

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

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

خطوة 2.1.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 0 & 1 & 7 \\ 0 & - 1 & 1 & - 2 & - 4 \\ 3 & 1 & 5 & - 7 & 1 \end{array}]$

خطوة 2.2

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

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خطوة 2.2.1

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

$[\begin{array}{c} 1 & 2 & 0 & 1 & 7 \\ 0 & - 1 & 1 & - 2 & - 4 \\ 3 - 3 \cdot 1 & 1 - 3 \cdot 2 & 5 - 3 \cdot 0 & - 7 - 3 \cdot 1 & 1 - 3 \cdot 7 \end{array}]$

خطوة 2.2.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 0 & 1 & 7 \\ 0 & - 1 & 1 & - 2 & - 4 \\ 0 & - 5 & 5 & - 10 & - 20 \end{array}]$

خطوة 2.3

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

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خطوة 2.3.1

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

$[\begin{array}{c} 1 & 2 & 0 & 1 & 7 \\ - 0 & - - 1 & - 1 \cdot 1 & - - 2 & - - 4 \\ 0 & - 5 & 5 & - 10 & - 20 \end{array}]$

خطوة 2.3.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 0 & 1 & 7 \\ 0 & 1 & - 1 & 2 & 4 \\ 0 & - 5 & 5 & - 10 & - 20 \end{array}]$

خطوة 2.4

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 2.4.1

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

$[\begin{array}{c} 1 & 2 & 0 & 1 & 7 \\ 0 & 1 & - 1 & 2 & 4 \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & 5 + 5 \cdot - 1 & - 10 + 5 \cdot 2 & - 20 + 5 \cdot 4 \end{array}]$

خطوة 2.4.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 0 & 1 & 7 \\ 0 & 1 & - 1 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

خطوة 2.5

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 2.5.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot - 1 & 1 - 2 \cdot 2 & 7 - 2 \cdot 4 \\ 0 & 1 & - 1 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

خطوة 2.5.2

بسّط $R_{1}$ .

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

خطوة 3

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

$a + 2 d - 3 g = - 1$

$b - d + 2 g = 4$

$0 = 0$

خطوة 4

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

$(- 1 - 2 d + 3 g, 4 + d - 2 g, d, g)$