Finite Mathematik Beispiele

$- 5 x + 5 y - 20 z = - 35$ , $x - y + 4 z = 7$ , $4 x - 4 y + 16 z = 28$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} - 5 & 5 & - 20 & - 35 \\ 1 & - 1 & 4 & 7 \\ 4 & - 4 & 16 & 28 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} - \frac{1}{5} \cdot - 5 & - \frac{1}{5} \cdot 5 & - \frac{1}{5} \cdot - 20 & - \frac{1}{5} \cdot - 35 \\ 1 & - 1 & 4 & 7 \\ 4 & - 4 & 16 & 28 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 4 & 7 \\ 1 & - 1 & 4 & 7 \\ 4 & - 4 & 16 & 28 \end{array}]$

Schritt 2.2

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

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Schritt 2.2.1

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

$[\begin{array}{c} 1 & - 1 & 4 & 7 \\ 1 - 1 & - 1 + 1 & 4 - 4 & 7 - 7 \\ 4 & - 4 & 16 & 28 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 4 & 7 \\ 0 & 0 & 0 & 0 \\ 4 & - 4 & 16 & 28 \end{array}]$

Schritt 2.3

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 2.3.1

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

$[\begin{array}{c} 1 & - 1 & 4 & 7 \\ 0 & 0 & 0 & 0 \\ 4 - 4 \cdot 1 & - 4 - 4 \cdot - 1 & 16 - 4 \cdot 4 & 28 - 4 \cdot 7 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{3}$ .

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

Schritt 3

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

$x - y + 4 z = 7$

$0 = 0$

Schritt 4

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

$(7 + y - 4 z, y, z)$