Finite Mathematik Beispiele

$x + 3 y - 8 z = 20$ , $y - 3 z = 11$ , $2 y + 7 z = - 4$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 3 & - 8 & 20 \\ 0 & 1 & - 3 & 11 \\ 0 & 2 & 7 & - 4 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} 1 & 3 & - 8 & 20 \\ 0 & 1 & - 3 & 11 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & 7 - 2 \cdot - 3 & - 4 - 2 \cdot 11 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - 8 & 20 \\ 0 & 1 & - 3 & 11 \\ 0 & 0 & 13 & - 26 \end{array}]$

Schritt 2.2

Multiply each element of $R_{3}$ by $\frac{1}{13}$ to make the entry at $3, 3$ a $1$ .

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Schritt 2.2.1

Multiply each element of $R_{3}$ by $\frac{1}{13}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 3 & - 8 & 20 \\ 0 & 1 & - 3 & 11 \\ \frac{0}{13} & \frac{0}{13} & \frac{13}{13} & \frac{- 26}{13} \end{array}]$

Schritt 2.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - 8 & 20 \\ 0 & 1 & - 3 & 11 \\ 0 & 0 & 1 & - 2 \end{array}]$

Schritt 2.3

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

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Schritt 2.3.1

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

$[\begin{array}{c} 1 & 3 & - 8 & 20 \\ 0 + 3 \cdot 0 & 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 11 + 3 \cdot - 2 \\ 0 & 0 & 1 & - 2 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - 8 & 20 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & - 2 \end{array}]$

Schritt 2.4

Perform the row operation $R_{1} = R_{1} + 8 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 2.4.1

Perform the row operation $R_{1} = R_{1} + 8 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 8 \cdot 0 & 3 + 8 \cdot 0 & - 8 + 8 \cdot 1 & 20 + 8 \cdot - 2 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & - 2 \end{array}]$

Schritt 2.4.2

Vereinfache $R_{1}$ .

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

Schritt 2.5

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

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Schritt 2.5.1

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

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

Schritt 2.5.2

Vereinfache $R_{1}$ .

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

Schritt 3

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

$x = - 11$

$y = 5$

$z = - 2$

Schritt 4

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

$(- 11, 5, - 2)$