Elementarmathematik Beispiele

$5 x - 3 y + 3 z = 58$ , $5 x - 4 y - 5 z = - 26$ , $x - y + 3 z = 34$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 5 & - 3 & 3 & 58 \\ 5 & - 4 & - 5 & - 26 \\ 1 & - 1 & 3 & 34 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} \frac{5}{5} & \frac{- 3}{5} & \frac{3}{5} & \frac{58}{5} \\ 5 & - 4 & - 5 & - 26 \\ 1 & - 1 & 3 & 34 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 5 & - 4 & - 5 & - 26 \\ 1 & - 1 & 3 & 34 \end{array}]$

Schritt 2.2

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 5 - 5 \cdot 1 & - 4 - 5 (- \frac{3}{5}) & - 5 - 5 (\frac{3}{5}) & - 26 - 5 (\frac{58}{5}) \\ 1 & - 1 & 3 & 34 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & - 1 & - 8 & - 84 \\ 1 & - 1 & 3 & 34 \end{array}]$

Schritt 2.3

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

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Schritt 2.3.1

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

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & - 1 & - 8 & - 84 \\ 1 - 1 & - 1 + \frac{3}{5} & 3 - \frac{3}{5} & 34 - \frac{58}{5} \end{array}]$

Schritt 2.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & - 1 & - 8 & - 84 \\ 0 & - \frac{2}{5} & \frac{12}{5} & \frac{112}{5} \end{array}]$

Schritt 2.4

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

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Schritt 2.4.1

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

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ - 0 & - - 1 & - - 8 & - - 84 \\ 0 & - \frac{2}{5} & \frac{12}{5} & \frac{112}{5} \end{array}]$

Schritt 2.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & 1 & 8 & 84 \\ 0 & - \frac{2}{5} & \frac{12}{5} & \frac{112}{5} \end{array}]$

Schritt 2.5

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

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Schritt 2.5.1

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

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & 1 & 8 & 84 \\ 0 + \frac{2}{5} \cdot 0 & - \frac{2}{5} + \frac{2}{5} \cdot 1 & \frac{12}{5} + \frac{2}{5} \cdot 8 & \frac{112}{5} + \frac{2}{5} \cdot 84 \end{array}]$

Schritt 2.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & 1 & 8 & 84 \\ 0 & 0 & \frac{28}{5} & 56 \end{array}]$

Schritt 2.6

Multiply each element of $R_{3}$ by $\frac{5}{28}$ to make the entry at $3, 3$ a $1$ .

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Schritt 2.6.1

Multiply each element of $R_{3}$ by $\frac{5}{28}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & 1 & 8 & 84 \\ \frac{5}{28} \cdot 0 & \frac{5}{28} \cdot 0 & \frac{5}{28} \cdot \frac{28}{5} & \frac{5}{28} \cdot 56 \end{array}]$

Schritt 2.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & 1 & 8 & 84 \\ 0 & 0 & 1 & 10 \end{array}]$

Schritt 2.7

Perform the row operation $R_{2} = R_{2} - 8 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 2.7.1

Perform the row operation $R_{2} = R_{2} - 8 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 - 8 \cdot 0 & 1 - 8 \cdot 0 & 8 - 8 \cdot 1 & 84 - 8 \cdot 10 \\ 0 & 0 & 1 & 10 \end{array}]$

Schritt 2.7.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & \frac{3}{5} & \frac{58}{5} \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 10 \end{array}]$

Schritt 2.8

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

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Schritt 2.8.1

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

$[\begin{array}{c} 1 - \frac{3}{5} \cdot 0 & - \frac{3}{5} - \frac{3}{5} \cdot 0 & \frac{3}{5} - \frac{3}{5} \cdot 1 & \frac{58}{5} - \frac{3}{5} \cdot 10 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 10 \end{array}]$

Schritt 2.8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{3}{5} & 0 & \frac{28}{5} \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 10 \end{array}]$

Schritt 2.9

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

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Schritt 2.9.1

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

$[\begin{array}{c} 1 + \frac{3}{5} \cdot 0 & - \frac{3}{5} + \frac{3}{5} \cdot 1 & 0 + \frac{3}{5} \cdot 0 & \frac{28}{5} + \frac{3}{5} \cdot 4 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 10 \end{array}]$

Schritt 2.9.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 8 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 10 \end{array}]$

Schritt 3

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

$x = 8$

$y = 4$

$z = 10$

Schritt 4

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

$(8, 4, 10)$