Finite Mathematik Beispiele

$x + y = 1$ , $5 x - 4 y = \frac{16}{5}$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 1 \\ 5 & - 4 & \frac{16}{5} \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.1.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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

Schritt 2.1.2

Vereinfache $R_{2}$ .

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

Schritt 2.2

Multiply each element of $R_{2}$ by $- \frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.2.1

Multiply each element of $R_{2}$ by $- \frac{1}{9}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 \\ - \frac{1}{9} \cdot 0 & - \frac{1}{9} \cdot - 9 & - \frac{1}{9} (- \frac{9}{5}) \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 \\ 0 & 1 & \frac{1}{5} \end{array}]$

Schritt 2.3

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

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Schritt 2.3.1

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

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

Schritt 2.3.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{4}{5} \\ 0 & 1 & \frac{1}{5} \end{array}]$

Schritt 3

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

$x = \frac{4}{5}$

$y = \frac{1}{5}$

Schritt 4

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

$(\frac{4}{5}, \frac{1}{5})$