Finite Mathematik Beispiele

$3 x + 2 y = - 6$ , $5 x - 2 y = - 10$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 3 & 2 & - 6 \\ 5 & - 2 & - 10 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{2}{3} & \frac{- 6}{3} \\ 5 & - 2 & - 10 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & - 2 \\ 5 & - 2 & - 10 \end{array}]$

Schritt 2.2

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 2.2.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & - 2 \\ 5 - 5 \cdot 1 & - 2 - 5 (\frac{2}{3}) & - 10 - 5 \cdot - 2 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & - 2 \\ 0 & - \frac{16}{3} & 0 \end{array}]$

Schritt 2.3

Multiply each element of $R_{2}$ by $- \frac{3}{16}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.3.1

Multiply each element of $R_{2}$ by $- \frac{3}{16}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & - 2 \\ - \frac{3}{16} \cdot 0 & - \frac{3}{16} (- \frac{16}{3}) & - \frac{3}{16} \cdot 0 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{2}$ .

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

Schritt 2.4

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

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Schritt 2.4.1

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

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

Schritt 2.4.2

Vereinfache $R_{1}$ .

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

Schritt 3

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

$x = - 2$

$y = 0$

Schritt 4

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

$(- 2, 0)$