Finite Mathematik Beispiele

$2 x + 3 y - 3 z = 0$ , $3 x - y + 4 z = 1$ , $5 x + 2 y + z = 1$

Schritt 1

Write the system as a matrix.

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

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{3}{2} & \frac{- 3}{2} & \frac{0}{2} \\ 3 & - 1 & 4 & 1 \\ 5 & 2 & 1 & 1 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ 3 & - 1 & 4 & 1 \\ 5 & 2 & 1 & 1 \end{array}]$

Schritt 2.2

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 2.2.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ 3 - 3 \cdot 1 & - 1 - 3 (\frac{3}{2}) & 4 - 3 (- \frac{3}{2}) & 1 - 3 \cdot 0 \\ 5 & 2 & 1 & 1 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{11}{2} & \frac{17}{2} & 1 \\ 5 & 2 & 1 & 1 \end{array}]$

Schritt 2.3

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

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Schritt 2.3.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{11}{2} & \frac{17}{2} & 1 \\ 5 - 5 \cdot 1 & 2 - 5 (\frac{3}{2}) & 1 - 5 (- \frac{3}{2}) & 1 - 5 \cdot 0 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{11}{2} & \frac{17}{2} & 1 \\ 0 & - \frac{11}{2} & \frac{17}{2} & 1 \end{array}]$

Schritt 2.4

Multiply each element of $R_{2}$ by $- \frac{2}{11}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.4.1

Multiply each element of $R_{2}$ by $- \frac{2}{11}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ - \frac{2}{11} \cdot 0 & - \frac{2}{11} (- \frac{11}{2}) & - \frac{2}{11} \cdot \frac{17}{2} & - \frac{2}{11} \cdot 1 \\ 0 & - \frac{11}{2} & \frac{17}{2} & 1 \end{array}]$

Schritt 2.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - \frac{17}{11} & - \frac{2}{11} \\ 0 & - \frac{11}{2} & \frac{17}{2} & 1 \end{array}]$

Schritt 2.5

Perform the row operation $R_{3} = R_{3} + \frac{11}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 2.5.1

Perform the row operation $R_{3} = R_{3} + \frac{11}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - \frac{17}{11} & - \frac{2}{11} \\ 0 + \frac{11}{2} \cdot 0 & - \frac{11}{2} + \frac{11}{2} \cdot 1 & \frac{17}{2} + \frac{11}{2} (- \frac{17}{11}) & 1 + \frac{11}{2} (- \frac{2}{11}) \end{array}]$

Schritt 2.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - \frac{17}{11} & - \frac{2}{11} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 2.6

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

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Schritt 2.6.1

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

$[\begin{array}{c} 1 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & - \frac{3}{2} - \frac{3}{2} (- \frac{17}{11}) & 0 - \frac{3}{2} (- \frac{2}{11}) \\ 0 & 1 & - \frac{17}{11} & - \frac{2}{11} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 2.6.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{9}{11} & \frac{3}{11} \\ 0 & 1 & - \frac{17}{11} & - \frac{2}{11} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 3

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

$x + \frac{9}{11} z = \frac{3}{11}$

$y - \frac{17}{11} z = - \frac{2}{11}$

$0 = 0$

Schritt 4

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

$(\frac{3}{11} - \frac{9 z}{11}, - \frac{2}{11} + \frac{17 z}{11}, z)$