Finite Mathematik Beispiele

$x + y + z + w = - 1$ , $3 x + 3 y - 3 z - 3 w = 9$ , $2 x - 3 y + 3 z + w = - 16$ , $x - y + 5 z + 2 w = - 16$

Schritt 1

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

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Schritt 1.1

Bewege $z$ .

$x + y + w + z = - 1$

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

$2 x - 3 y + 3 z + w = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.2

Bewege $y$ .

$x + w + y + z = - 1$

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

$2 x - 3 y + 3 z + w = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.3

Stelle $x$ und $w$ um.

$w + x + y + z = - 1$

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

$2 x - 3 y + 3 z + w = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.4

Bewege $- 3 z$ .

$w + x + y + z = - 1$

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

$2 x - 3 y + 3 z + w = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.5

Bewege $3 y$ .

$w + x + y + z = - 1$

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

$2 x - 3 y + 3 z + w = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.6

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

$w + x + y + z = - 1$

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

$2 x - 3 y + 3 z + w = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.7

Bewege $3 z$ .

$w + x + y + z = - 1$

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

$2 x - 3 y + w + 3 z = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.8

Bewege $- 3 y$ .

$w + x + y + z = - 1$

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

$2 x + w - 3 y + 3 z = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.9

Stelle $2 x$ und $w$ um.

$w + x + y + z = - 1$

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

$w + 2 x - 3 y + 3 z = - 16$

$x - y + 5 z + 2 w = - 16$

Schritt 1.10

Bewege $5 z$ .

$w + x + y + z = - 1$

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

$w + 2 x - 3 y + 3 z = - 16$

$x - y + 2 w + 5 z = - 16$

Schritt 1.11

Bewege $- y$ .

$w + x + y + z = - 1$

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

$w + 2 x - 3 y + 3 z = - 16$

$x + 2 w - y + 5 z = - 16$

Schritt 1.12

Stelle $x$ und $2 w$ um.

$w + x + y + z = - 1$

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

$w + 2 x - 3 y + 3 z = - 16$

$2 w + x - y + 5 z = - 16$

$w + x + y + z = - 1$

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

$w + 2 x - 3 y + 3 z = - 16$

$2 w + x - y + 5 z = - 16$

Schritt 2

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ - 3 & 3 & 3 & - 3 & 9 \\ 1 & 2 & - 3 & 3 & - 16 \\ 2 & 1 & - 1 & 5 & - 16 \end{array}]$

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

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 3.1.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ - 3 + 3 \cdot 1 & 3 + 3 \cdot 1 & 3 + 3 \cdot 1 & - 3 + 3 \cdot 1 & 9 + 3 \cdot - 1 \\ 1 & 2 & - 3 & 3 & - 16 \\ 2 & 1 & - 1 & 5 & - 16 \end{array}]$

Schritt 3.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 6 & 6 & 0 & 6 \\ 1 & 2 & - 3 & 3 & - 16 \\ 2 & 1 & - 1 & 5 & - 16 \end{array}]$

Schritt 3.2

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

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Schritt 3.2.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 6 & 6 & 0 & 6 \\ 1 - 1 & 2 - 1 & - 3 - 1 & 3 - 1 & - 16 + 1 \\ 2 & 1 & - 1 & 5 & - 16 \end{array}]$

Schritt 3.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 6 & 6 & 0 & 6 \\ 0 & 1 & - 4 & 2 & - 15 \\ 2 & 1 & - 1 & 5 & - 16 \end{array}]$

Schritt 3.3

Perform the row operation $R_{4} = R_{4} - 2 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Schritt 3.3.1

Perform the row operation $R_{4} = R_{4} - 2 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 6 & 6 & 0 & 6 \\ 0 & 1 & - 4 & 2 & - 15 \\ 2 - 2 \cdot 1 & 1 - 2 \cdot 1 & - 1 - 2 \cdot 1 & 5 - 2 \cdot 1 & - 16 - 2 \cdot - 1 \end{array}]$

Schritt 3.3.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 6 & 6 & 0 & 6 \\ 0 & 1 & - 4 & 2 & - 15 \\ 0 & - 1 & - 3 & 3 & - 14 \end{array}]$

Schritt 3.4

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Schritt 3.4.1

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ \frac{0}{6} & \frac{6}{6} & \frac{6}{6} & \frac{0}{6} & \frac{6}{6} \\ 0 & 1 & - 4 & 2 & - 15 \\ 0 & - 1 & - 3 & 3 & - 14 \end{array}]$

Schritt 3.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & - 4 & 2 & - 15 \\ 0 & - 1 & - 3 & 3 & - 14 \end{array}]$

Schritt 3.5

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

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Schritt 3.5.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 - 0 & 1 - 1 & - 4 - 1 & 2 - 0 & - 15 - 1 \\ 0 & - 1 & - 3 & 3 & - 14 \end{array}]$

Schritt 3.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & - 5 & 2 & - 16 \\ 0 & - 1 & - 3 & 3 & - 14 \end{array}]$

Schritt 3.6

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

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Schritt 3.6.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & - 5 & 2 & - 16 \\ 0 + 0 & - 1 + 1 \cdot 1 & - 3 + 1 \cdot 1 & 3 + 0 & - 14 + 1 \cdot 1 \end{array}]$

Schritt 3.6.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & - 5 & 2 & - 16 \\ 0 & 0 & - 2 & 3 & - 13 \end{array}]$

Schritt 3.7

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

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Schritt 3.7.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ - \frac{1}{5} \cdot 0 & - \frac{1}{5} \cdot 0 & - \frac{1}{5} \cdot - 5 & - \frac{1}{5} \cdot 2 & - \frac{1}{5} \cdot - 16 \\ 0 & 0 & - 2 & 3 & - 13 \end{array}]$

Schritt 3.7.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & - \frac{2}{5} & \frac{16}{5} \\ 0 & 0 & - 2 & 3 & - 13 \end{array}]$

Schritt 3.8

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

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Schritt 3.8.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & - \frac{2}{5} & \frac{16}{5} \\ 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 3 + 2 (- \frac{2}{5}) & - 13 + 2 (\frac{16}{5}) \end{array}]$

Schritt 3.8.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & - \frac{2}{5} & \frac{16}{5} \\ 0 & 0 & 0 & \frac{11}{5} & - \frac{33}{5} \end{array}]$

Schritt 3.9

Multiply each element of $R_{4}$ by $\frac{5}{11}$ to make the entry at $4, 4$ a $1$ .

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Schritt 3.9.1

Multiply each element of $R_{4}$ by $\frac{5}{11}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & - \frac{2}{5} & \frac{16}{5} \\ \frac{5}{11} \cdot 0 & \frac{5}{11} \cdot 0 & \frac{5}{11} \cdot 0 & \frac{5}{11} \cdot \frac{11}{5} & \frac{5}{11} (- \frac{33}{5}) \end{array}]$

Schritt 3.9.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & - \frac{2}{5} & \frac{16}{5} \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Schritt 3.10

Perform the row operation $R_{3} = R_{3} + \frac{2}{5} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Schritt 3.10.1

Perform the row operation $R_{3} = R_{3} + \frac{2}{5} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 \\ 0 & 1 & 1 & 0 & 1 \\ 0 + \frac{2}{5} \cdot 0 & 0 + \frac{2}{5} \cdot 0 & 1 + \frac{2}{5} \cdot 0 & - \frac{2}{5} + \frac{2}{5} \cdot 1 & \frac{16}{5} + \frac{2}{5} \cdot - 3 \\ 0 & 0 & 0 & 1 & - 3 \end{array}]$

Schritt 3.10.2

Vereinfache $R_{3}$ .

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

Schritt 3.11

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

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Schritt 3.11.1

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

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

Schritt 3.11.2

Vereinfache $R_{1}$ .

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

Schritt 3.12

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

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Schritt 3.12.1

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

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

Schritt 3.12.2

Vereinfache $R_{2}$ .

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

Schritt 3.13

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

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Schritt 3.13.1

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

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

Schritt 3.13.2

Vereinfache $R_{1}$ .

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

Schritt 3.14

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

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Schritt 3.14.1

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

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

Schritt 3.14.2

Vereinfache $R_{1}$ .

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

Schritt 4

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

$w = 1$

$x = - 1$

$y = 2$

$z = - 3$

Schritt 5

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

$(1, - 1, 2, - 3)$