Finite Mathematik Beispiele

$4 x - 8 y = 4$ , $- 8 x + 16 y = - 8$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 4 & - 8 & 4 \\ - 8 & 16 & - 8 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{4}{4} & \frac{- 8}{4} & \frac{4}{4} \\ - 8 & 16 & - 8 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & 1 \\ - 8 & 16 & - 8 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - 2 & 1 \\ - 8 + 8 \cdot 1 & 16 + 8 \cdot - 2 & - 8 + 8 \cdot 1 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

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

Schritt 3

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

$x - 2 y = 1$

$0 = 0$

Schritt 4

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

$(1 + 2 y, y)$