Finite Mathematik Beispiele

$- 30 x - 9 y + 18 z = 30$ , $10 x + 3 y - 6 z = - 10$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} - 30 & - 9 & 18 & 30 \\ 10 & 3 & - 6 & - 10 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1

Multiply each element of $R_{1}$ by $- \frac{1}{30}$ to make the entry at $1, 1$ a $1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.1

Multiply each element of $R_{1}$ by $- \frac{1}{30}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{30} \cdot - 30 & - \frac{1}{30} \cdot - 9 & - \frac{1}{30} \cdot 18 & - \frac{1}{30} \cdot 30 \\ 10 & 3 & - 6 & - 10 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

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

Schritt 2.2

Perform the row operation $R_{2} = R_{2} - 10 R_{1}$ to make the entry at $2, 1$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.2.1

Perform the row operation $R_{2} = R_{2} - 10 R_{1}$ to make the entry at $2, 1$ a $0$ .

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

Schritt 2.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{10} & - \frac{3}{5} & - 1 \\ 0 & 0 & 0 & 0 \end{array}]$

Schritt 3

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

$x + \frac{3}{10} y - \frac{3}{5} z = - 1$

$0 = 0$

Schritt 4

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

$(- 1 - \frac{3 y}{10} + \frac{3 z}{5}, y, z)$