Elementarmathematik Beispiele

$x - 3 y - z = - 9$ , $- x + 8 y - 4 z = 24$ , $2 x - 15 y + 7 z = - 45$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ - 1 & 8 & - 4 & 24 \\ 2 & - 15 & 7 & - 45 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ - 1 + 1 \cdot 1 & 8 - 3 & - 4 - 1 & 24 - 9 \\ 2 & - 15 & 7 & - 45 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 5 & - 5 & 15 \\ 2 & - 15 & 7 & - 45 \end{array}]$

Schritt 2.2

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 2.2.1

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

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 5 & - 5 & 15 \\ 2 - 2 \cdot 1 & - 15 - 2 \cdot - 3 & 7 - 2 \cdot - 1 & - 45 - 2 \cdot - 9 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 5 & - 5 & 15 \\ 0 & - 9 & 9 & - 27 \end{array}]$

Schritt 2.3

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ \frac{0}{5} & \frac{5}{5} & \frac{- 5}{5} & \frac{15}{5} \\ 0 & - 9 & 9 & - 27 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 1 & - 1 & 3 \\ 0 & - 9 & 9 & - 27 \end{array}]$

Schritt 2.4

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

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Schritt 2.4.1

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

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 1 & - 1 & 3 \\ 0 + 9 \cdot 0 & - 9 + 9 \cdot 1 & 9 + 9 \cdot - 1 & - 27 + 9 \cdot 3 \end{array}]$

Schritt 2.4.2

Vereinfache $R_{3}$ .

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

Schritt 2.5

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 2.5.1

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

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

Schritt 2.5.2

Vereinfache $R_{1}$ .

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

Schritt 3

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

$x - 4 z = 0$

$y - z = 3$

$0 = 0$

Schritt 4

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

$(4 z, 3 + z, z)$