Finite Mathematik Beispiele

$x - 2 y + 3 z + w = - 14$ , $x - 3 y + z - w = - 25$ , $x - y = - 8$ , $x + z + w = 2$

Schritt 1

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

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Schritt 1.1

Bewege $3 z$ .

$x - 2 y + w + 3 z = - 14$

$x - 3 y + z - w = - 25$

$x - y = - 8$

$x + z + w = 2$

Schritt 1.2

Bewege $- 2 y$ .

$x + w - 2 y + 3 z = - 14$

$x - 3 y + z - w = - 25$

$x - y = - 8$

$x + z + w = 2$

Schritt 1.3

Stelle $x$ und $w$ um.

$w + x - 2 y + 3 z = - 14$

$x - 3 y + z - w = - 25$

$x - y = - 8$

$x + z + w = 2$

Schritt 1.4

Bewege $z$ .

$w + x - 2 y + 3 z = - 14$

$x - 3 y - w + z = - 25$

$x - y = - 8$

$x + z + w = 2$

Schritt 1.5

Bewege $- 3 y$ .

$w + x - 2 y + 3 z = - 14$

$x - w - 3 y + z = - 25$

$x - y = - 8$

$x + z + w = 2$

Schritt 1.6

Stelle $x$ und $- w$ um.

$w + x - 2 y + 3 z = - 14$

$- w + x - 3 y + z = - 25$

$x - y = - 8$

$x + z + w = 2$

Schritt 1.7

Bewege $z$ .

$w + x - 2 y + 3 z = - 14$

$- w + x - 3 y + z = - 25$

$x - y = - 8$

$x + w + z = 2$

Schritt 1.8

Stelle $x$ und $w$ um.

$w + x - 2 y + 3 z = - 14$

$- w + x - 3 y + z = - 25$

$x - y = - 8$

$w + x + z = 2$

$w + x - 2 y + 3 z = - 14$

$- w + x - 3 y + z = - 25$

$x - y = - 8$

$w + x + z = 2$

Schritt 2

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ - 1 & 1 & - 3 & 1 & - 25 \\ 0 & 1 & - 1 & 0 & - 8 \\ 1 & 1 & 0 & 1 & 2 \end{array}]$

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

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

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Schritt 3.1.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ - 1 + 1 \cdot 1 & 1 + 1 \cdot 1 & - 3 - 2 & 1 + 1 \cdot 3 & - 25 - 14 \\ 0 & 1 & - 1 & 0 & - 8 \\ 1 & 1 & 0 & 1 & 2 \end{array}]$

Schritt 3.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 2 & - 5 & 4 & - 39 \\ 0 & 1 & - 1 & 0 & - 8 \\ 1 & 1 & 0 & 1 & 2 \end{array}]$

Schritt 3.2

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

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Schritt 3.2.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 2 & - 5 & 4 & - 39 \\ 0 & 1 & - 1 & 0 & - 8 \\ 1 - 1 & 1 - 1 & 0 + 2 & 1 - 3 & 2 + 14 \end{array}]$

Schritt 3.2.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 2 & - 5 & 4 & - 39 \\ 0 & 1 & - 1 & 0 & - 8 \\ 0 & 0 & 2 & - 2 & 16 \end{array}]$

Schritt 3.3

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

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Schritt 3.3.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ \frac{0}{2} & \frac{2}{2} & \frac{- 5}{2} & \frac{4}{2} & \frac{- 39}{2} \\ 0 & 1 & - 1 & 0 & - 8 \\ 0 & 0 & 2 & - 2 & 16 \end{array}]$

Schritt 3.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 & 1 & - 1 & 0 & - 8 \\ 0 & 0 & 2 & - 2 & 16 \end{array}]$

Schritt 3.4

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

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Schritt 3.4.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 - 0 & 1 - 1 & - 1 + \frac{5}{2} & 0 - 2 & - 8 + \frac{39}{2} \\ 0 & 0 & 2 & - 2 & 16 \end{array}]$

Schritt 3.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 & 0 & \frac{3}{2} & - 2 & \frac{23}{2} \\ 0 & 0 & 2 & - 2 & 16 \end{array}]$

Schritt 3.5

Multiply each element of $R_{3}$ by $\frac{2}{3}$ to make the entry at $3, 3$ a $1$ .

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Schritt 3.5.1

Multiply each element of $R_{3}$ by $\frac{2}{3}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ \frac{2}{3} \cdot 0 & \frac{2}{3} \cdot 0 & \frac{2}{3} \cdot \frac{3}{2} & \frac{2}{3} \cdot - 2 & \frac{2}{3} \cdot \frac{23}{2} \\ 0 & 0 & 2 & - 2 & 16 \end{array}]$

Schritt 3.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 & 0 & 1 & - \frac{4}{3} & \frac{23}{3} \\ 0 & 0 & 2 & - 2 & 16 \end{array}]$

Schritt 3.6

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 3.6.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 & 0 & 1 & - \frac{4}{3} & \frac{23}{3} \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 2 - 2 (- \frac{4}{3}) & 16 - 2 (\frac{23}{3}) \end{array}]$

Schritt 3.6.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 & 0 & 1 & - \frac{4}{3} & \frac{23}{3} \\ 0 & 0 & 0 & \frac{2}{3} & \frac{2}{3} \end{array}]$

Schritt 3.7

Multiply each element of $R_{4}$ by $\frac{3}{2}$ to make the entry at $4, 4$ a $1$ .

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Schritt 3.7.1

Multiply each element of $R_{4}$ by $\frac{3}{2}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 & 0 & 1 & - \frac{4}{3} & \frac{23}{3} \\ \frac{3}{2} \cdot 0 & \frac{3}{2} \cdot 0 & \frac{3}{2} \cdot 0 & \frac{3}{2} \cdot \frac{2}{3} & \frac{3}{2} \cdot \frac{2}{3} \end{array}]$

Schritt 3.7.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 & 0 & 1 & - \frac{4}{3} & \frac{23}{3} \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.8

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

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Schritt 3.8.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 + \frac{4}{3} \cdot 0 & 0 + \frac{4}{3} \cdot 0 & 1 + \frac{4}{3} \cdot 0 & - \frac{4}{3} + \frac{4}{3} \cdot 1 & \frac{23}{3} + \frac{4}{3} \cdot 1 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.8.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 2 & - \frac{39}{2} \\ 0 & 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.9

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

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Schritt 3.9.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & - \frac{5}{2} - 2 \cdot 0 & 2 - 2 \cdot 1 & - \frac{39}{2} - 2 \cdot 1 \\ 0 & 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.9.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 3 & - 14 \\ 0 & 1 & - \frac{5}{2} & 0 & - \frac{43}{2} \\ 0 & 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.10

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

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Schritt 3.10.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 1 - 3 \cdot 0 & - 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & - 14 - 3 \cdot 1 \\ 0 & 1 & - \frac{5}{2} & 0 & - \frac{43}{2} \\ 0 & 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.10.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 17 \\ 0 & 1 & - \frac{5}{2} & 0 & - \frac{43}{2} \\ 0 & 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.11

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

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Schritt 3.11.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 17 \\ 0 + \frac{5}{2} \cdot 0 & 1 + \frac{5}{2} \cdot 0 & - \frac{5}{2} + \frac{5}{2} \cdot 1 & 0 + \frac{5}{2} \cdot 0 & - \frac{43}{2} + \frac{5}{2} \cdot 9 \\ 0 & 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.11.2

Vereinfache $R_{2}$ .

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

Schritt 3.12

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 3.12.1

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

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

Schritt 3.12.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 3.13

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

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Schritt 3.13.1

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

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

Schritt 3.13.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 9 \\ 0 & 0 & 0 & 1 & 1 \end{array}]$

Schritt 4

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

$w = 0$

$x = 1$

$y = 9$

$z = 1$

Schritt 5

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

$(0, 1, 9, 1)$