Finite Mathematik Beispiele

$2 x - 5 y + 3 z = 9$ , $2 x + 4 y - z = 13$ , $6 x - 6 y + 5 z = 31$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 2 & - 5 & 3 & 9 \\ 2 & 4 & - 1 & 13 \\ 6 & - 6 & 5 & 31 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{- 5}{2} & \frac{3}{2} & \frac{9}{2} \\ 2 & 4 & - 1 & 13 \\ 6 & - 6 & 5 & 31 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ 2 & 4 & - 1 & 13 \\ 6 & - 6 & 5 & 31 \end{array}]$

Schritt 2.2

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

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Schritt 2.2.1

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

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ 2 - 2 \cdot 1 & 4 - 2 (- \frac{5}{2}) & - 1 - 2 (\frac{3}{2}) & 13 - 2 (\frac{9}{2}) \\ 6 & - 6 & 5 & 31 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ 0 & 9 & - 4 & 4 \\ 6 & - 6 & 5 & 31 \end{array}]$

Schritt 2.3

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 2.3.1

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ 0 & 9 & - 4 & 4 \\ 6 - 6 \cdot 1 & - 6 - 6 (- \frac{5}{2}) & 5 - 6 (\frac{3}{2}) & 31 - 6 (\frac{9}{2}) \end{array}]$

Schritt 2.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ 0 & 9 & - 4 & 4 \\ 0 & 9 & - 4 & 4 \end{array}]$

Schritt 2.4

Multiply each element of $R_{2}$ by $\frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ \frac{0}{9} & \frac{9}{9} & \frac{- 4}{9} & \frac{4}{9} \\ 0 & 9 & - 4 & 4 \end{array}]$

Schritt 2.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ 0 & 1 & - \frac{4}{9} & \frac{4}{9} \\ 0 & 9 & - 4 & 4 \end{array}]$

Schritt 2.5

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 2.5.1

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

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ 0 & 1 & - \frac{4}{9} & \frac{4}{9} \\ 0 - 9 \cdot 0 & 9 - 9 \cdot 1 & - 4 - 9 (- \frac{4}{9}) & 4 - 9 (\frac{4}{9}) \end{array}]$

Schritt 2.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{2} & \frac{3}{2} & \frac{9}{2} \\ 0 & 1 & - \frac{4}{9} & \frac{4}{9} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 2.6

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

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Schritt 2.6.1

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

$[\begin{array}{c} 1 + \frac{5}{2} \cdot 0 & - \frac{5}{2} + \frac{5}{2} \cdot 1 & \frac{3}{2} + \frac{5}{2} (- \frac{4}{9}) & \frac{9}{2} + \frac{5}{2} \cdot \frac{4}{9} \\ 0 & 1 & - \frac{4}{9} & \frac{4}{9} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 2.6.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{7}{18} & \frac{101}{18} \\ 0 & 1 & - \frac{4}{9} & \frac{4}{9} \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 3

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

$x + \frac{7}{18} z = \frac{101}{18}$

$y - \frac{4}{9} z = \frac{4}{9}$

$0 = 0$

Schritt 4

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

$(\frac{101}{18} - \frac{7 z}{18}, \frac{4}{9} + \frac{4 z}{9}, z)$