Finite Mathematik Beispiele

$x - 2 y = 8$ , $3 x + 4 y = 4$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 2 & 8 \\ 3 & 4 & 4 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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 2.1.1

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

$[\begin{array}{c} 1 & - 2 & 8 \\ 3 - 3 \cdot 1 & 4 - 3 \cdot - 2 & 4 - 3 \cdot 8 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 8 \\ 0 & 10 & - 20 \end{array}]$

Schritt 2.2

Multiply each element of $R_{2}$ by $\frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.2.1

Multiply each element of $R_{2}$ by $\frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 8 \\ \frac{0}{10} & \frac{10}{10} & \frac{- 20}{10} \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

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

Schritt 2.3

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

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Schritt 2.3.1

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

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

Schritt 2.3.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 4 \\ 0 & 1 & - 2 \end{array}]$

Schritt 3

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

$x = 4$

$y = - 2$

Schritt 4

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

$(4, - 2)$