Algebra Beispiele

$x_{1} + 6 x_{2} - 2 x_{3} = 10$ $3 x_{1} + 8 x_{3} = - 4$ $- 2 x_{1} + 5 x_{2} + 3 x_{3} = 22$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 3 & 0 & 8 & - 4 \\ - 2 & 5 & 3 & 22 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 3 - 3 \cdot 1 & 0 - 3 \cdot 6 & 8 - 3 \cdot - 2 & - 4 - 3 \cdot 10 \\ - 2 & 5 & 3 & 22 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & - 18 & 14 & - 34 \\ - 2 & 5 & 3 & 22 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & - 18 & 14 & - 34 \\ - 2 + 2 \cdot 1 & 5 + 2 \cdot 6 & 3 + 2 \cdot - 2 & 22 + 2 \cdot 10 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & - 18 & 14 & - 34 \\ 0 & 17 & - 1 & 42 \end{array}]$

Schritt 2.3

Multiply each element of $R_{2}$ by $- \frac{1}{18}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{18}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ - \frac{1}{18} \cdot 0 & - \frac{1}{18} \cdot - 18 & - \frac{1}{18} \cdot 14 & - \frac{1}{18} \cdot - 34 \\ 0 & 17 & - 1 & 42 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & 1 & - \frac{7}{9} & \frac{17}{9} \\ 0 & 17 & - 1 & 42 \end{array}]$

Schritt 2.4

Perform the row operation $R_{3} = R_{3} - 17 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 2.4.1

Perform the row operation $R_{3} = R_{3} - 17 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & 1 & - \frac{7}{9} & \frac{17}{9} \\ 0 - 17 \cdot 0 & 17 - 17 \cdot 1 & - 1 - 17 (- \frac{7}{9}) & 42 - 17 (\frac{17}{9}) \end{array}]$

Schritt 2.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & 1 & - \frac{7}{9} & \frac{17}{9} \\ 0 & 0 & \frac{110}{9} & \frac{89}{9} \end{array}]$

Schritt 2.5

Multiply each element of $R_{3}$ by $\frac{9}{110}$ to make the entry at $3, 3$ a $1$ .

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Schritt 2.5.1

Multiply each element of $R_{3}$ by $\frac{9}{110}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & 1 & - \frac{7}{9} & \frac{17}{9} \\ \frac{9}{110} \cdot 0 & \frac{9}{110} \cdot 0 & \frac{9}{110} \cdot \frac{110}{9} & \frac{9}{110} \cdot \frac{89}{9} \end{array}]$

Schritt 2.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & 1 & - \frac{7}{9} & \frac{17}{9} \\ 0 & 0 & 1 & \frac{89}{110} \end{array}]$

Schritt 2.6

Perform the row operation $R_{2} = R_{2} + \frac{7}{9} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 2.6.1

Perform the row operation $R_{2} = R_{2} + \frac{7}{9} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 + \frac{7}{9} \cdot 0 & 1 + \frac{7}{9} \cdot 0 & - \frac{7}{9} + \frac{7}{9} \cdot 1 & \frac{17}{9} + \frac{7}{9} \cdot \frac{89}{110} \\ 0 & 0 & 1 & \frac{89}{110} \end{array}]$

Schritt 2.6.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 6 & - 2 & 10 \\ 0 & 1 & 0 & \frac{277}{110} \\ 0 & 0 & 1 & \frac{89}{110} \end{array}]$

Schritt 2.7

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 2.7.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & 6 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 10 + 2 (\frac{89}{110}) \\ 0 & 1 & 0 & \frac{277}{110} \\ 0 & 0 & 1 & \frac{89}{110} \end{array}]$

Schritt 2.7.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 6 & 0 & \frac{639}{55} \\ 0 & 1 & 0 & \frac{277}{110} \\ 0 & 0 & 1 & \frac{89}{110} \end{array}]$

Schritt 2.8

Perform the row operation $R_{1} = R_{1} - 6 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 2.8.1

Perform the row operation $R_{1} = R_{1} - 6 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 6 \cdot 0 & 6 - 6 \cdot 1 & 0 - 6 \cdot 0 & \frac{639}{55} - 6 (\frac{277}{110}) \\ 0 & 1 & 0 & \frac{277}{110} \\ 0 & 0 & 1 & \frac{89}{110} \end{array}]$

Schritt 2.8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{192}{55} \\ 0 & 1 & 0 & \frac{277}{110} \\ 0 & 0 & 1 & \frac{89}{110} \end{array}]$

Schritt 3

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

$x_{1} = - \frac{192}{55}$

$x_{2} = \frac{277}{110}$

$x_{3} = \frac{89}{110}$

Schritt 4

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

$(- \frac{192}{55}, \frac{277}{110}, \frac{89}{110})$