الرياضيات المتناهية الأمثلة

$x - y + z - u + v = - 1$ , $- x + y + u - v = 2$ , $z - 9 v = 1$ , $x - y + 4 z - u - 4 v = 2$ , $x - y + 8 z - u - 9 v = 6$

خطوة 1

Move variables to the left and constant terms to the right.

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خطوة 1.1

انقُل $z$ .

$x - y - u + v + z = - 1$

$- x + y + u - v = 2$

$z - 9 v = 1$

$x - y + 4 z - u - 4 v = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.2

انقُل $- y$ .

$x - u + v - y + z = - 1$

$- x + y + u - v = 2$

$z - 9 v = 1$

$x - y + 4 z - u - 4 v = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.3

انقُل $x$ .

$- u + v + x - y + z = - 1$

$- x + y + u - v = 2$

$z - 9 v = 1$

$x - y + 4 z - u - 4 v = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.4

انقُل $y$ .

$- u + v + x - y + z = - 1$

$- x + u - v + y = 2$

$z - 9 v = 1$

$x - y + 4 z - u - 4 v = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.5

انقُل $- x$ .

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$z - 9 v = 1$

$x - y + 4 z - u - 4 v = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.6

أعِد ترتيب $z$ و $- 9 v$ .

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$- 9 v + z = 1$

$x - y + 4 z - u - 4 v = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.7

انقُل $4 z$ .

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$- 9 v + z = 1$

$x - y - u - 4 v + 4 z = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.8

انقُل $- y$ .

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$- 9 v + z = 1$

$x - u - 4 v - y + 4 z = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.9

انقُل $x$ .

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$- 9 v + z = 1$

$- u - 4 v + x - y + 4 z = 2$

$x - y + 8 z - u - 9 v = 6$

خطوة 1.10

انقُل $8 z$ .

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$- 9 v + z = 1$

$- u - 4 v + x - y + 4 z = 2$

$x - y - u - 9 v + 8 z = 6$

خطوة 1.11

انقُل $- y$ .

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$- 9 v + z = 1$

$- u - 4 v + x - y + 4 z = 2$

$x - u - 9 v - y + 8 z = 6$

خطوة 1.12

انقُل $x$ .

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$- 9 v + z = 1$

$- u - 4 v + x - y + 4 z = 2$

$- u - 9 v + x - y + 8 z = 6$

$- u + v + x - y + z = - 1$

$u - v - x + y = 2$

$- 9 v + z = 1$

$- u - 4 v + x - y + 4 z = 2$

$- u - 9 v + x - y + 8 z = 6$

خطوة 2

Write the system as a matrix.

$[\begin{array}{c} - 1 & 1 & 1 & - 1 & 1 & - 1 \\ 1 & - 1 & - 1 & 1 & 0 & 2 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ - 1 & - 4 & 1 & - 1 & 4 & 2 \\ - 1 & - 9 & 1 & - 1 & 8 & 6 \end{array}]$

خطوة 3

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 3.1

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

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خطوة 3.1.1

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

$[\begin{array}{c} - - 1 & - 1 \cdot 1 & - 1 \cdot 1 & - - 1 & - 1 \cdot 1 & - - 1 \\ 1 & - 1 & - 1 & 1 & 0 & 2 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ - 1 & - 4 & 1 & - 1 & 4 & 2 \\ - 1 & - 9 & 1 & - 1 & 8 & 6 \end{array}]$

خطوة 3.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 1 & - 1 & - 1 & 1 & 0 & 2 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ - 1 & - 4 & 1 & - 1 & 4 & 2 \\ - 1 & - 9 & 1 & - 1 & 8 & 6 \end{array}]$

خطوة 3.2

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

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خطوة 3.2.1

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

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 1 - 1 & - 1 + 1 & - 1 + 1 & 1 - 1 & 0 + 1 & 2 - 1 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ - 1 & - 4 & 1 & - 1 & 4 & 2 \\ - 1 & - 9 & 1 & - 1 & 8 & 6 \end{array}]$

خطوة 3.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ - 1 & - 4 & 1 & - 1 & 4 & 2 \\ - 1 & - 9 & 1 & - 1 & 8 & 6 \end{array}]$

خطوة 3.3

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

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خطوة 3.3.1

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

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ - 1 + 1 \cdot 1 & - 4 - 1 & 1 - 1 & - 1 + 1 \cdot 1 & 4 - 1 & 2 + 1 \cdot 1 \\ - 1 & - 9 & 1 & - 1 & 8 & 6 \end{array}]$

خطوة 3.3.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ 0 & - 5 & 0 & 0 & 3 & 3 \\ - 1 & - 9 & 1 & - 1 & 8 & 6 \end{array}]$

خطوة 3.4

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

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خطوة 3.4.1

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

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ 0 & - 5 & 0 & 0 & 3 & 3 \\ - 1 + 1 \cdot 1 & - 9 - 1 & 1 - 1 & - 1 + 1 \cdot 1 & 8 - 1 & 6 + 1 \cdot 1 \end{array}]$

خطوة 3.4.2

بسّط $R_{5}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ 0 & - 5 & 0 & 0 & 3 & 3 \\ 0 & - 10 & 0 & 0 & 7 & 7 \end{array}]$

خطوة 3.5

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & - 9 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & - 5 & 0 & 0 & 3 & 3 \\ 0 & - 10 & 0 & 0 & 7 & 7 \end{array}]$

خطوة 3.6

Multiply each element of $R_{2}$ by $- \frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

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خطوة 3.6.1

Multiply each element of $R_{2}$ by $- \frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ - \frac{1}{9} \cdot 0 & - \frac{1}{9} \cdot - 9 & - \frac{1}{9} \cdot 0 & - \frac{1}{9} \cdot 0 & - \frac{1}{9} \cdot 1 & - \frac{1}{9} \cdot 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & - 5 & 0 & 0 & 3 & 3 \\ 0 & - 10 & 0 & 0 & 7 & 7 \end{array}]$

خطوة 3.6.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & - 5 & 0 & 0 & 3 & 3 \\ 0 & - 10 & 0 & 0 & 7 & 7 \end{array}]$

خطوة 3.7

Perform the row operation $R_{4} = R_{4} + 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

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خطوة 3.7.1

Perform the row operation $R_{4} = R_{4} + 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & 0 + 5 \cdot 0 & 0 + 5 \cdot 0 & 3 + 5 (- \frac{1}{9}) & 3 + 5 (- \frac{1}{9}) \\ 0 & - 10 & 0 & 0 & 7 & 7 \end{array}]$

خطوة 3.7.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & \frac{22}{9} & \frac{22}{9} \\ 0 & - 10 & 0 & 0 & 7 & 7 \end{array}]$

خطوة 3.8

Perform the row operation $R_{5} = R_{5} + 10 R_{2}$ to make the entry at $5, 2$ a $0$ .

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خطوة 3.8.1

Perform the row operation $R_{5} = R_{5} + 10 R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & \frac{22}{9} & \frac{22}{9} \\ 0 + 10 \cdot 0 & - 10 + 10 \cdot 1 & 0 + 10 \cdot 0 & 0 + 10 \cdot 0 & 7 + 10 (- \frac{1}{9}) & 7 + 10 (- \frac{1}{9}) \end{array}]$

خطوة 3.8.2

بسّط $R_{5}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & \frac{22}{9} & \frac{22}{9} \\ 0 & 0 & 0 & 0 & \frac{53}{9} & \frac{53}{9} \end{array}]$

خطوة 3.9

Perform the row operation $R_{4} = R_{4} - \frac{22}{9} R_{3}$ to make the entry at $4, 5$ a $0$ .

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خطوة 3.9.1

Perform the row operation $R_{4} = R_{4} - \frac{22}{9} R_{3}$ to make the entry at $4, 5$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 - \frac{22}{9} \cdot 0 & 0 - \frac{22}{9} \cdot 0 & 0 - \frac{22}{9} \cdot 0 & 0 - \frac{22}{9} \cdot 0 & \frac{22}{9} - \frac{22}{9} \cdot 1 & \frac{22}{9} - \frac{22}{9} \cdot 1 \\ 0 & 0 & 0 & 0 & \frac{53}{9} & \frac{53}{9} \end{array}]$

خطوة 3.9.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{53}{9} & \frac{53}{9} \end{array}]$

خطوة 3.10

Perform the row operation $R_{5} = R_{5} - \frac{53}{9} R_{3}$ to make the entry at $5, 5$ a $0$ .

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خطوة 3.10.1

Perform the row operation $R_{5} = R_{5} - \frac{53}{9} R_{3}$ to make the entry at $5, 5$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 - \frac{53}{9} \cdot 0 & 0 - \frac{53}{9} \cdot 0 & 0 - \frac{53}{9} \cdot 0 & 0 - \frac{53}{9} \cdot 0 & \frac{53}{9} - \frac{53}{9} \cdot 1 & \frac{53}{9} - \frac{53}{9} \cdot 1 \end{array}]$

خطوة 3.10.2

بسّط $R_{5}$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 & 1 & 0 & 0 & - \frac{1}{9} & - \frac{1}{9} \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

خطوة 3.11

Perform the row operation $R_{2} = R_{2} + \frac{1}{9} R_{3}$ to make the entry at $2, 5$ a $0$ .

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خطوة 3.11.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{9} R_{3}$ to make the entry at $2, 5$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 0 + \frac{1}{9} \cdot 0 & 1 + \frac{1}{9} \cdot 0 & 0 + \frac{1}{9} \cdot 0 & 0 + \frac{1}{9} \cdot 0 & - \frac{1}{9} + \frac{1}{9} \cdot 1 & - \frac{1}{9} + \frac{1}{9} \cdot 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

خطوة 3.11.2

بسّط $R_{2}$ .

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

خطوة 3.12

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

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خطوة 3.12.1

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

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

خطوة 3.12.2

بسّط $R_{1}$ .

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

خطوة 3.13

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

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خطوة 3.13.1

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

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

خطوة 3.13.2

بسّط $R_{1}$ .

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

خطوة 4

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

$u - x + y = 2$

$v = 0$

$z = 1$

$0 = 0$

خطوة 5

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

$(2 + x - y, 0, x, y, 1)$