Finite Mathematik Beispiele

$4 x - 5 y = 15$ , $8 x - 10 y = 30$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 4 & - 5 & 15 \\ 8 & - 10 & 30 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

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Schritt 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{4}{4} & \frac{- 5}{4} & \frac{15}{4} \\ 8 & - 10 & 30 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{5}{4} & \frac{15}{4} \\ 8 & - 10 & 30 \end{array}]$

Schritt 2.2

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.2.1

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{4} & \frac{15}{4} \\ 8 - 8 \cdot 1 & - 10 - 8 (- \frac{5}{4}) & 30 - 8 (\frac{15}{4}) \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{4} & \frac{15}{4} \\ 0 & 0 & 0 \end{array}]$

Schritt 3

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

$x - \frac{5}{4} y = \frac{15}{4}$

$0 = 0$

Schritt 4

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

$(\frac{15}{4} + \frac{5 y}{4}, y)$