Finite Mathematik Beispiele

$x - 2 y + 3 z = 9$ , $y - z + w = - 9$ , $- 3 x + 3 y - 3 z + 4 w = 5$ , $2 y - 3 z + w = - 9$

Schritt 1

Move variables to the left and constant terms to the right.

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Schritt 1.1

Bewege $- z$ .

$x - 2 y + 3 z = 9$

$y + w - z = - 9$

$- 3 x + 3 y - 3 z + 4 w = 5$

$2 y - 3 z + w = - 9$

Schritt 1.2

Stelle $y$ und $w$ um.

$x - 2 y + 3 z = 9$

$w + y - z = - 9$

$- 3 x + 3 y - 3 z + 4 w = 5$

$2 y - 3 z + w = - 9$

Schritt 1.3

Bewege $- 3 z$ .

$x - 2 y + 3 z = 9$

$w + y - z = - 9$

$- 3 x + 3 y + 4 w - 3 z = 5$

$2 y - 3 z + w = - 9$

Schritt 1.4

Bewege $3 y$ .

$x - 2 y + 3 z = 9$

$w + y - z = - 9$

$- 3 x + 4 w + 3 y - 3 z = 5$

$2 y - 3 z + w = - 9$

Schritt 1.5

Stelle $- 3 x$ und $4 w$ um.

$x - 2 y + 3 z = 9$

$w + y - z = - 9$

$4 w - 3 x + 3 y - 3 z = 5$

$2 y - 3 z + w = - 9$

Schritt 1.6

Bewege $- 3 z$ .

$x - 2 y + 3 z = 9$

$w + y - z = - 9$

$4 w - 3 x + 3 y - 3 z = 5$

$2 y + w - 3 z = - 9$

Schritt 1.7

Stelle $2 y$ und $w$ um.

$x - 2 y + 3 z = 9$

$w + y - z = - 9$

$4 w - 3 x + 3 y - 3 z = 5$

$w + 2 y - 3 z = - 9$

$x - 2 y + 3 z = 9$

$w + y - z = - 9$

$4 w - 3 x + 3 y - 3 z = 5$

$w + 2 y - 3 z = - 9$

Schritt 2

Write the system as a matrix.

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

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

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

Schritt 3.2

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 3.2.1

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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

Schritt 3.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & - 3 & - 1 & 1 & 41 \\ 1 & 0 & 2 & - 3 & - 9 \end{array}]$

Schritt 3.3

Perform the row operation $R_{4} = R_{4} - R_{1}$ to make the entry at $4, 1$ a $0$ .

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Schritt 3.3.1

Perform the row operation $R_{4} = R_{4} - R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & - 3 & - 1 & 1 & 41 \\ 1 - 1 & 0 - 0 & 2 - 1 & - 3 + 1 & - 9 + 9 \end{array}]$

Schritt 3.3.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & - 3 & - 1 & 1 & 41 \\ 0 & 0 & 1 & - 2 & 0 \end{array}]$

Schritt 3.4

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

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Schritt 3.4.1

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

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 1 + 3 \cdot - 2 & 1 + 3 \cdot 3 & 41 + 3 \cdot 9 \\ 0 & 0 & 1 & - 2 & 0 \end{array}]$

Schritt 3.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & 0 & - 7 & 10 & 68 \\ 0 & 0 & 1 & - 2 & 0 \end{array}]$

Schritt 3.5

Multiply each element of $R_{3}$ by $- \frac{1}{7}$ to make the entry at $3, 3$ a $1$ .

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Schritt 3.5.1

Multiply each element of $R_{3}$ by $- \frac{1}{7}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ - \frac{1}{7} \cdot 0 & - \frac{1}{7} \cdot 0 & - \frac{1}{7} \cdot - 7 & - \frac{1}{7} \cdot 10 & - \frac{1}{7} \cdot 68 \\ 0 & 0 & 1 & - 2 & 0 \end{array}]$

Schritt 3.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{68}{7} \\ 0 & 0 & 1 & - 2 & 0 \end{array}]$

Schritt 3.6

Perform the row operation $R_{4} = R_{4} - R_{3}$ to make the entry at $4, 3$ a $0$ .

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Schritt 3.6.1

Perform the row operation $R_{4} = R_{4} - R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{68}{7} \\ 0 - 0 & 0 - 0 & 1 - 1 & - 2 + \frac{10}{7} & 0 + \frac{68}{7} \end{array}]$

Schritt 3.6.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{68}{7} \\ 0 & 0 & 0 & - \frac{4}{7} & \frac{68}{7} \end{array}]$

Schritt 3.7

Multiply each element of $R_{4}$ by $- \frac{7}{4}$ to make the entry at $4, 4$ a $1$ .

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Schritt 3.7.1

Multiply each element of $R_{4}$ by $- \frac{7}{4}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{68}{7} \\ - \frac{7}{4} \cdot 0 & - \frac{7}{4} \cdot 0 & - \frac{7}{4} \cdot 0 & - \frac{7}{4} (- \frac{4}{7}) & - \frac{7}{4} \cdot \frac{68}{7} \end{array}]$

Schritt 3.7.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{68}{7} \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.8

Perform the row operation $R_{3} = R_{3} + \frac{10}{7} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Schritt 3.8.1

Perform the row operation $R_{3} = R_{3} + \frac{10}{7} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 + \frac{10}{7} \cdot 0 & 0 + \frac{10}{7} \cdot 0 & 1 + \frac{10}{7} \cdot 0 & - \frac{10}{7} + \frac{10}{7} \cdot 1 & - \frac{68}{7} + \frac{10}{7} \cdot - 17 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.8.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 3 & 9 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.9

Perform the row operation $R_{2} = R_{2} - 3 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Schritt 3.9.1

Perform the row operation $R_{2} = R_{2} - 3 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 - 3 \cdot 0 & 1 - 3 \cdot 0 & - 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & 9 - 3 \cdot - 17 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.9.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 9 \\ 0 & 1 & - 2 & 0 & 60 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.10

Perform the row operation $R_{1} = R_{1} + R_{4}$ to make the entry at $1, 4$ a $0$ .

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Schritt 3.10.1

Perform the row operation $R_{1} = R_{1} + R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + 0 & 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & - 9 - 17 \\ 0 & 1 & - 2 & 0 & 60 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.10.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 26 \\ 0 & 1 & - 2 & 0 & 60 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.11

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

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Schritt 3.11.1

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

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 26 \\ 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & 60 + 2 \cdot - 34 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.11.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 26 \\ 0 & 1 & 0 & 0 & - 8 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.12

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

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Schritt 3.12.1

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

$[\begin{array}{c} 1 - 0 & 0 - 0 & 1 - 1 & 0 - 0 & - 26 + 34 \\ 0 & 1 & 0 & 0 & - 8 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 3.12.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 8 \\ 0 & 1 & 0 & 0 & - 8 \\ 0 & 0 & 1 & 0 & - 34 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Schritt 4

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

$w = 8$

$x = - 8$

$y = - 34$

$z = - 17$

Schritt 5

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

$(8, - 8, - 34, - 17)$