Finite Mathematik Beispiele

$y = 30 - 6 x$ , $6 x + y = 60$

Schritt 1

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

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Schritt 1.1

Addiere $6 x$ zu beiden Seiten der Gleichung.

$y + 6 x = 30$

$6 x + y = 60$

Schritt 1.2

Stelle $y$ und $6 x$ um.

$6 x + y = 30$

$6 x + y = 60$

$6 x + y = 30$

$6 x + y = 60$

Schritt 2

Write the system as a matrix.

$[\begin{array}{c} 6 & 1 & 30 \\ 6 & 1 & 60 \end{array}]$

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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Schritt 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{6}{6} & \frac{1}{6} & \frac{30}{6} \\ 6 & 1 & 60 \end{array}]$

Schritt 3.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{6} & 5 \\ 6 & 1 & 60 \end{array}]$

Schritt 3.2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 3.2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{6} & 5 \\ 6 - 6 \cdot 1 & 1 - 6 (\frac{1}{6}) & 60 - 6 \cdot 5 \end{array}]$

Schritt 3.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{6} & 5 \\ 0 & 0 & 30 \end{array}]$

Schritt 3.3

Multiply each element of $R_{2}$ by $\frac{1}{30}$ to make the entry at $2, 3$ a $1$ .

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Schritt 3.3.1

Multiply each element of $R_{2}$ by $\frac{1}{30}$ to make the entry at $2, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{6} & 5 \\ \frac{0}{30} & \frac{0}{30} & \frac{30}{30} \end{array}]$

Schritt 3.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{6} & 5 \\ 0 & 0 & 1 \end{array}]$

Schritt 3.4

Perform the row operation $R_{1} = R_{1} - 5 R_{2}$ to make the entry at $1, 3$ a $0$ .

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Schritt 3.4.1

Perform the row operation $R_{1} = R_{1} - 5 R_{2}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 5 \cdot 0 & \frac{1}{6} - 5 \cdot 0 & 5 - 5 \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Schritt 3.4.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{6} & 0 \\ 0 & 0 & 1 \end{array}]$

Schritt 4

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

$x + \frac{1}{6} y = 0$

$0 = 1$

Schritt 5

The system is inconsistent so there is no solution.

$Keine Lösung$