Lineare Algebra Beispiele

$- a + 5 b = 12$ , $- a - 3 b = 10$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} - 1 & 5 & 12 \\ - 1 & - 3 & 10 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} - - 1 & - 1 \cdot 5 & - 1 \cdot 12 \\ - 1 & - 3 & 10 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - 5 & - 12 \\ - 1 & - 3 & 10 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - 5 & - 12 \\ - 1 + 1 \cdot 1 & - 3 - 5 & 10 - 12 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 5 & - 12 \\ 0 & - 8 & - 2 \end{array}]$

Schritt 2.3

Multiply each element of $R_{2}$ by $- \frac{1}{8}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{8}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 5 & - 12 \\ - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot - 8 & - \frac{1}{8} \cdot - 2 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 5 & - 12 \\ 0 & 1 & \frac{1}{4} \end{array}]$

Schritt 2.4

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

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Schritt 2.4.1

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

$[\begin{array}{c} 1 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - 12 + 5 (\frac{1}{4}) \\ 0 & 1 & \frac{1}{4} \end{array}]$

Schritt 2.4.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{43}{4} \\ 0 & 1 & \frac{1}{4} \end{array}]$

Schritt 3

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

$a = - \frac{43}{4}$

$b = \frac{1}{4}$

Schritt 4

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

$(- \frac{43}{4}, \frac{1}{4})$