Elementarmathematik Beispiele

$x + y + 4 z = 1$ , $3 x + 2 y + 11 z = 1$ , $x + 3 z = - 1$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 4 & 1 \\ 3 & 2 & 11 & 1 \\ 1 & 0 & 3 & - 1 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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 2.1.1

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

$[\begin{array}{c} 1 & 1 & 4 & 1 \\ 3 - 3 \cdot 1 & 2 - 3 \cdot 1 & 11 - 3 \cdot 4 & 1 - 3 \cdot 1 \\ 1 & 0 & 3 & - 1 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{2}$ .

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

Schritt 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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Schritt 2.2.1

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

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

Schritt 2.2.2

Vereinfache $R_{3}$ .

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

Schritt 2.3

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

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Schritt 2.3.1

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

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

Schritt 2.3.2

Vereinfache $R_{2}$ .

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

Schritt 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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Schritt 2.4.1

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

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

Schritt 2.4.2

Vereinfache $R_{3}$ .

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

Schritt 2.5

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

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Schritt 2.5.1

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

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

Schritt 2.5.2

Vereinfache $R_{1}$ .

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

Schritt 3

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

$x + 3 z = - 1$

$y + z = 2$

$0 = 0$

Schritt 4

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

$(- 1 - 3 z, 2 - z, z)$