Elementarmathematik Beispiele

$2 x + 3 y = 2$ , $4 x + 6 y = 4$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 2 & 3 & 2 \\ 4 & 6 & 4 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{3}{2} & \frac{2}{2} \\ 4 & 6 & 4 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & 1 \\ 4 & 6 & 4 \end{array}]$

Schritt 2.2

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.2.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{2} & 1 \\ 4 - 4 \cdot 1 & 6 - 4 (\frac{3}{2}) & 4 - 4 \cdot 1 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{2} & 1 \\ 0 & 0 & 0 \end{array}]$

Schritt 3

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

$x + \frac{3}{2} y = 1$

$0 = 0$

Schritt 4

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

$(1 - \frac{3 y}{2}, y)$