Finite Mathematik Beispiele

$2 y = 14$ , $x = y + 1$

Schritt 1

Subtrahiere $y$ von beiden Seiten der Gleichung.

$2 y = 14$

$x - y = 1$

Schritt 2

Write the system as a matrix.

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

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

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

Schritt 3.2

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

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Schritt 3.2.1

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

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

Schritt 3.2.2

Vereinfache $R_{2}$ .

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

Schritt 3.3

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

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Schritt 3.3.1

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

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

Schritt 3.3.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 8 \\ 0 & 1 & 7 \end{array}]$

Schritt 4

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

$x = 8$

$y = 7$

Schritt 5

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

$(8, 7)$