الرياضيات المتناهية الأمثلة

$x + y = 103$ , $20 x + 10 y = 1500$ , $x = 45$ , $y = 32$

خطوة 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 103 \\ 20 & 10 & 1500 \\ 1 & 0 & 45 \\ 0 & 1 & 32 \end{array}]$

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 2.1

Perform the row operation $R_{2} = R_{2} - 20 R_{1}$ to make the entry at $2, 1$ a $0$ .

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خطوة 2.1.1

Perform the row operation $R_{2} = R_{2} - 20 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 20 - 20 \cdot 1 & 10 - 20 \cdot 1 & 1500 - 20 \cdot 103 \\ 1 & 0 & 45 \\ 0 & 1 & 32 \end{array}]$

خطوة 2.1.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & - 10 & - 560 \\ 1 & 0 & 45 \\ 0 & 1 & 32 \end{array}]$

خطوة 2.2

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

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خطوة 2.2.1

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

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & - 10 & - 560 \\ 1 - 1 & 0 - 1 & 45 - 103 \\ 0 & 1 & 32 \end{array}]$

خطوة 2.2.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & - 10 & - 560 \\ 0 & - 1 & - 58 \\ 0 & 1 & 32 \end{array}]$

خطوة 2.3

Multiply each element of $R_{2}$ by $- \frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

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خطوة 2.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 103 \\ - \frac{1}{10} \cdot 0 & - \frac{1}{10} \cdot - 10 & - \frac{1}{10} \cdot - 560 \\ 0 & - 1 & - 58 \\ 0 & 1 & 32 \end{array}]$

خطوة 2.3.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ 0 & - 1 & - 58 \\ 0 & 1 & 32 \end{array}]$

خطوة 2.4

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

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خطوة 2.4.1

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

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ 0 + 0 & - 1 + 1 \cdot 1 & - 58 + 1 \cdot 56 \\ 0 & 1 & 32 \end{array}]$

خطوة 2.4.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ 0 & 0 & - 2 \\ 0 & 1 & 32 \end{array}]$

خطوة 2.5

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

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خطوة 2.5.1

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

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ 0 & 0 & - 2 \\ 0 - 0 & 1 - 1 & 32 - 56 \end{array}]$

خطوة 2.5.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ 0 & 0 & - 2 \\ 0 & 0 & - 24 \end{array}]$

خطوة 2.6

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

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خطوة 2.6.1

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

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 \\ 0 & 0 & - 24 \end{array}]$

خطوة 2.6.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ 0 & 0 & 1 \\ 0 & 0 & - 24 \end{array}]$

خطوة 2.7

Perform the row operation $R_{4} = R_{4} + 24 R_{3}$ to make the entry at $4, 3$ a $0$ .

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خطوة 2.7.1

Perform the row operation $R_{4} = R_{4} + 24 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ 0 & 0 & 1 \\ 0 + 24 \cdot 0 & 0 + 24 \cdot 0 & - 24 + 24 \cdot 1 \end{array}]$

خطوة 2.7.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 56 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}]$

خطوة 2.8

Perform the row operation $R_{2} = R_{2} - 56 R_{3}$ to make the entry at $2, 3$ a $0$ .

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خطوة 2.8.1

Perform the row operation $R_{2} = R_{2} - 56 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 - 56 \cdot 0 & 1 - 56 \cdot 0 & 56 - 56 \cdot 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}]$

خطوة 2.8.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 103 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}]$

خطوة 2.9

Perform the row operation $R_{1} = R_{1} - 103 R_{3}$ to make the entry at $1, 3$ a $0$ .

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خطوة 2.9.1

Perform the row operation $R_{1} = R_{1} - 103 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 103 \cdot 0 & 1 - 103 \cdot 0 & 103 - 103 \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}]$

خطوة 2.9.2

بسّط $R_{1}$ .

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

خطوة 2.10

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

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خطوة 2.10.1

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

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

خطوة 2.10.2

بسّط $R_{1}$ .

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

خطوة 3

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

$x = 0$

$y = 0$

$0 = 1$

$0 = 0$

خطوة 4

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

$(0, 0)$