Finite Mathematik Beispiele

$3 x - 9 y + 6 z = - 12$ , $x - 3 y + 2 x = - 4$ , $2 x - 6 y + 4 z = 8$

Schritt 1

Addiere $x$ und $2 x$ .

$3 x - 9 y + 6 z = - 12$

$3 x - 3 y = - 4$

$2 x - 6 y + 4 z = 8$

Schritt 2

Write the system as a matrix.

$[\begin{array}{c} 3 & - 9 & 6 & - 12 \\ 3 & - 3 & 0 & - 4 \\ 2 & - 6 & 4 & 8 \end{array}]$

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

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

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Schritt 3.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 9}{3} & \frac{6}{3} & \frac{- 12}{3} \\ 3 & - 3 & 0 & - 4 \\ 2 & - 6 & 4 & 8 \end{array}]$

Schritt 3.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 4 \\ 3 & - 3 & 0 & - 4 \\ 2 & - 6 & 4 & 8 \end{array}]$

Schritt 3.2

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 3.2.1

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

$[\begin{array}{c} 1 & - 3 & 2 & - 4 \\ 3 - 3 \cdot 1 & - 3 - 3 \cdot - 3 & 0 - 3 \cdot 2 & - 4 - 3 \cdot - 4 \\ 2 & - 6 & 4 & 8 \end{array}]$

Schritt 3.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 4 \\ 0 & 6 & - 6 & 8 \\ 2 & - 6 & 4 & 8 \end{array}]$

Schritt 3.3

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

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Schritt 3.3.1

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

$[\begin{array}{c} 1 & - 3 & 2 & - 4 \\ 0 & 6 & - 6 & 8 \\ 2 - 2 \cdot 1 & - 6 - 2 \cdot - 3 & 4 - 2 \cdot 2 & 8 - 2 \cdot - 4 \end{array}]$

Schritt 3.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 4 \\ 0 & 6 & - 6 & 8 \\ 0 & 0 & 0 & 16 \end{array}]$

Schritt 3.4

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Schritt 3.4.1

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 4 \\ \frac{0}{6} & \frac{6}{6} & \frac{- 6}{6} & \frac{8}{6} \\ 0 & 0 & 0 & 16 \end{array}]$

Schritt 3.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 4 \\ 0 & 1 & - 1 & \frac{4}{3} \\ 0 & 0 & 0 & 16 \end{array}]$

Schritt 3.5

Multiply each element of $R_{3}$ by $\frac{1}{16}$ to make the entry at $3, 4$ a $1$ .

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Schritt 3.5.1

Multiply each element of $R_{3}$ by $\frac{1}{16}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & - 3 & 2 & - 4 \\ 0 & 1 & - 1 & \frac{4}{3} \\ \frac{0}{16} & \frac{0}{16} & \frac{0}{16} & \frac{16}{16} \end{array}]$

Schritt 3.5.2

Vereinfache $R_{3}$ .

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

Schritt 3.6

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

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Schritt 3.6.1

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

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

Schritt 3.6.2

Vereinfache $R_{2}$ .

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

Schritt 3.7

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

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Schritt 3.7.1

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

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

Schritt 3.7.2

Vereinfache $R_{1}$ .

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

Schritt 3.8

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

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Schritt 3.8.1

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

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

Schritt 3.8.2

Vereinfache $R_{1}$ .

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

Schritt 4

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

$x - z = 0$

$y - z = 0$

$0 = 1$

Schritt 5

The system is inconsistent so there is no solution.

$Keine Lösung$