Finite Mathematik Beispiele

$x - 3 y = - 6$ , $5 x + 3 y = 42$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 3 & - 6 \\ 5 & 3 & 42 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - 3 & - 6 \\ 5 - 5 \cdot 1 & 3 - 5 \cdot - 3 & 42 - 5 \cdot - 6 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & - 6 \\ 0 & 18 & 72 \end{array}]$

Schritt 2.2

Multiply each element of $R_{2}$ by $\frac{1}{18}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.2.1

Multiply each element of $R_{2}$ by $\frac{1}{18}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 3 & - 6 \\ \frac{0}{18} & \frac{18}{18} & \frac{72}{18} \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & - 6 \\ 0 & 1 & 4 \end{array}]$

Schritt 2.3

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

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Schritt 2.3.1

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

$[\begin{array}{c} 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 6 + 3 \cdot 4 \\ 0 & 1 & 4 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 6 \\ 0 & 1 & 4 \end{array}]$

Schritt 3

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

$x = 6$

$y = 4$

Schritt 4

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

$(6, 4)$