Finite Mathematik Beispiele

$v = 3 i + j$ , $w = i + 3 j$

Schritt 1

Move variables to the left and constant terms to the right.

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Schritt 1.1

Subtrahiere $j$ von beiden Seiten der Gleichung.

$v - j = 3 i$

$w = i + 3 j$

Schritt 1.2

Stelle $v$ und $- j$ um.

$- j + v = 3 i$

$w = i + 3 j$

Schritt 1.3

Subtrahiere $3 j$ von beiden Seiten der Gleichung.

$- j + v = 3 i$

$w - 3 j = i$

Schritt 1.4

Stelle $w$ und $- 3 j$ um.

$- j + v = 3 i$

$- 3 j + w = i$

$- j + v = 3 i$

$- 3 j + w = i$

Schritt 2

Write the system as a matrix.

$[\begin{array}{c} - 1 & 1 & 0 & 3 i \\ - 3 & 0 & 1 & i \end{array}]$

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

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

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Schritt 3.1.1

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

$[\begin{array}{c} - - 1 & - 1 \cdot 1 & - 0 & - (3 i) \\ - 3 & 0 & 1 & i \end{array}]$

Schritt 3.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 3 i \\ - 3 & 0 & 1 & i \end{array}]$

Schritt 3.2

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

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Schritt 3.2.1

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

$[\begin{array}{c} 1 & - 1 & 0 & - 3 i \\ - 3 + 3 \cdot 1 & 0 + 3 \cdot - 1 & 1 + 3 \cdot 0 & i + 3 (- 3 i) \end{array}]$

Schritt 3.2.2

Vereinfache $R_{2}$ .

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

Schritt 3.3

Multiply each element of $R_{2}$ by $- \frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

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Schritt 3.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 3 i \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 1 & - \frac{1}{3} (- 8 i) \end{array}]$

Schritt 3.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 3 i \\ 0 & 1 & - \frac{1}{3} & \frac{8 i}{3} \end{array}]$

Schritt 3.4

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

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Schritt 3.4.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 0 - \frac{1}{3} & - 3 i + \frac{8 i}{3} \\ 0 & 1 & - \frac{1}{3} & \frac{8 i}{3} \end{array}]$

Schritt 3.4.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{1}{3} & - \frac{i}{3} \\ 0 & 1 & - \frac{1}{3} & \frac{8 i}{3} \end{array}]$

Schritt 4

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

$j - \frac{1}{3} w = - \frac{i}{3}$

$v - \frac{1}{3} w = \frac{8 i}{3}$

Schritt 5

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

$(- \frac{i}{3} + \frac{w}{3}, \frac{8 i}{3} + \frac{w}{3}, w)$