Finite Mathematik Beispiele

$3 x + y + 6 z = 32$ , $3 x + 2 y + 9 z = 46$ , $3 x + y + 5 z = 34$ , $3 x + 3 y + 3 z = 24$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 3 & 1 & 6 & 32 \\ 3 & 2 & 9 & 46 \\ 3 & 1 & 5 & 34 \\ 3 & 3 & 3 & 24 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{1}{3} & \frac{6}{3} & \frac{32}{3} \\ 3 & 2 & 9 & 46 \\ 3 & 1 & 5 & 34 \\ 3 & 3 & 3 & 24 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 3 & 2 & 9 & 46 \\ 3 & 1 & 5 & 34 \\ 3 & 3 & 3 & 24 \end{array}]$

Schritt 2.2

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

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Schritt 2.2.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 3 - 3 \cdot 1 & 2 - 3 (\frac{1}{3}) & 9 - 3 \cdot 2 & 46 - 3 (\frac{32}{3}) \\ 3 & 1 & 5 & 34 \\ 3 & 3 & 3 & 24 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 3 & 1 & 5 & 34 \\ 3 & 3 & 3 & 24 \end{array}]$

Schritt 2.3

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

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Schritt 2.3.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 3 - 3 \cdot 1 & 1 - 3 (\frac{1}{3}) & 5 - 3 \cdot 2 & 34 - 3 (\frac{32}{3}) \\ 3 & 3 & 3 & 24 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & - 1 & 2 \\ 3 & 3 & 3 & 24 \end{array}]$

Schritt 2.4

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Schritt 2.4.1

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & - 1 & 2 \\ 3 - 3 \cdot 1 & 3 - 3 (\frac{1}{3}) & 3 - 3 \cdot 2 & 24 - 3 (\frac{32}{3}) \end{array}]$

Schritt 2.4.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & - 1 & 2 \\ 0 & 2 & - 3 & - 8 \end{array}]$

Schritt 2.5

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

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Schritt 2.5.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & - 1 & 2 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 3 - 2 \cdot 3 & - 8 - 2 \cdot 14 \end{array}]$

Schritt 2.5.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & - 1 & 2 \\ 0 & 0 & - 9 & - 36 \end{array}]$

Schritt 2.6

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

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Schritt 2.6.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ - 0 & - 0 & - - 1 & - 1 \cdot 2 \\ 0 & 0 & - 9 & - 36 \end{array}]$

Schritt 2.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & 1 & - 2 \\ 0 & 0 & - 9 & - 36 \end{array}]$

Schritt 2.7

Perform the row operation $R_{4} = R_{4} + 9 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Schritt 2.7.1

Perform the row operation $R_{4} = R_{4} + 9 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & 1 & - 2 \\ 0 + 9 \cdot 0 & 0 + 9 \cdot 0 & - 9 + 9 \cdot 1 & - 36 + 9 \cdot - 2 \end{array}]$

Schritt 2.7.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & - 54 \end{array}]$

Schritt 2.8

Multiply each element of $R_{4}$ by $- \frac{1}{54}$ to make the entry at $4, 4$ a $1$ .

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Schritt 2.8.1

Multiply each element of $R_{4}$ by $- \frac{1}{54}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & 1 & - 2 \\ - \frac{1}{54} \cdot 0 & - \frac{1}{54} \cdot 0 & - \frac{1}{54} \cdot 0 & - \frac{1}{54} \cdot - 54 \end{array}]$

Schritt 2.8.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 2.9

Perform the row operation $R_{3} = R_{3} + 2 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Schritt 2.9.1

Perform the row operation $R_{3} = R_{3} + 2 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 2.9.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 14 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 2.10

Perform the row operation $R_{2} = R_{2} - 14 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Schritt 2.10.1

Perform the row operation $R_{2} = R_{2} - 14 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 - 14 \cdot 0 & 1 - 14 \cdot 0 & 3 - 14 \cdot 0 & 14 - 14 \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 2.10.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 2 & \frac{32}{3} \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 2.11

Perform the row operation $R_{1} = R_{1} - \frac{32}{3} R_{4}$ to make the entry at $1, 4$ a $0$ .

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Schritt 2.11.1

Perform the row operation $R_{1} = R_{1} - \frac{32}{3} R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{32}{3} \cdot 0 & \frac{1}{3} - \frac{32}{3} \cdot 0 & 2 - \frac{32}{3} \cdot 0 & \frac{32}{3} - \frac{32}{3} \cdot 1 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 2.11.2

Vereinfache $R_{1}$ .

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

Schritt 2.12

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

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Schritt 2.12.1

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

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

Schritt 2.12.2

Vereinfache $R_{2}$ .

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

Schritt 2.13

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

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Schritt 2.13.1

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

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

Schritt 2.13.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Schritt 2.14

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

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Schritt 2.14.1

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

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

Schritt 2.14.2

Vereinfache $R_{1}$ .

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

Schritt 3

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

$x = 0$

$y = 0$

$z = 0$

$0 = 1$

Schritt 4

The system is inconsistent so there is no solution.

$Keine Lösung$