Finite Mathematik Beispiele

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

Schritt 1

Write the system as a matrix.

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

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 $1$ .

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

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{3} & \frac{5}{3} \\ 6 & 10 & 10 \end{array}]$

Schritt 2.2

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 2.2.1

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

$[\begin{array}{c} 1 & \frac{5}{3} & \frac{5}{3} \\ 6 - 6 \cdot 1 & 10 - 6 (\frac{5}{3}) & 10 - 6 (\frac{5}{3}) \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{3} & \frac{5}{3} \\ 0 & 0 & 0 \end{array}]$

Schritt 3

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

$x + \frac{5}{3} y = \frac{5}{3}$

$0 = 0$

Schritt 4

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

$(\frac{5}{3} - \frac{5 y}{3}, y)$