Finite Mathematik Beispiele

$- 25 - 9 y - z = - 16$ , $- 15 - 15 y - 2 z = 26$ , $20 x + 9 y + z = 6$

Schritt 1

Move variables to the left and constant terms to the right.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1

Bringe alle Terme, die keine Variable enthalten, auf die rechte Seite der Gleichung.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1.1

Addiere $25$ zu beiden Seiten der Gleichung.

$- 9 y - z = - 16 + 25$

$- 15 - 15 y - 2 z = 26$

$20 x + 9 y + z = 6$

Schritt 1.1.2

Addiere $- 16$ und $25$ .

$- 9 y - z = 9$

$- 15 - 15 y - 2 z = 26$

$20 x + 9 y + z = 6$

$- 9 y - z = 9$

$- 15 - 15 y - 2 z = 26$

$20 x + 9 y + z = 6$

Schritt 1.2

Bringe alle Terme, die keine Variable enthalten, auf die rechte Seite der Gleichung.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.2.1

Addiere $15$ zu beiden Seiten der Gleichung.

$- 9 y - z = 9$

$- 15 y - 2 z = 26 + 15$

$20 x + 9 y + z = 6$

Schritt 1.2.2

Addiere $26$ und $15$ .

$- 9 y - z = 9$

$- 15 y - 2 z = 41$

$20 x + 9 y + z = 6$

$- 9 y - z = 9$

$- 15 y - 2 z = 41$

$20 x + 9 y + z = 6$

$- 9 y - z = 9$

$- 15 y - 2 z = 41$

$20 x + 9 y + z = 6$

Schritt 2

Write the system as a matrix.

$[\begin{array}{c} 0 & - 9 & - 1 & 9 \\ 0 & - 15 & - 2 & 41 \\ 20 & 9 & 1 & 6 \end{array}]$

Schritt 3

Ermittele die normierte Zeilenstufenform.

Tippen, um mehr Schritte zu sehen ...

Schritt 3.1

Swap $R_{3}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 20 & 9 & 1 & 6 \\ 0 & - 15 & - 2 & 41 \\ 0 & - 9 & - 1 & 9 \end{array}]$

Schritt 3.2

Multiply each element of $R_{1}$ by $\frac{1}{20}$ to make the entry at $1, 1$ a $1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{20}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{20}{20} & \frac{9}{20} & \frac{1}{20} & \frac{6}{20} \\ 0 & - 15 & - 2 & 41 \\ 0 & - 9 & - 1 & 9 \end{array}]$

Schritt 3.2.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ 0 & - 15 & - 2 & 41 \\ 0 & - 9 & - 1 & 9 \end{array}]$

Schritt 3.3

Multiply each element of $R_{2}$ by $- \frac{1}{15}$ to make the entry at $2, 2$ a $1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{15}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ - \frac{1}{15} \cdot 0 & - \frac{1}{15} \cdot - 15 & - \frac{1}{15} \cdot - 2 & - \frac{1}{15} \cdot 41 \\ 0 & - 9 & - 1 & 9 \end{array}]$

Schritt 3.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ 0 & 1 & \frac{2}{15} & - \frac{41}{15} \\ 0 & - 9 & - 1 & 9 \end{array}]$

Schritt 3.4

Perform the row operation $R_{3} = R_{3} + 9 R_{2}$ to make the entry at $3, 2$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.4.1

Perform the row operation $R_{3} = R_{3} + 9 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ 0 & 1 & \frac{2}{15} & - \frac{41}{15} \\ 0 + 9 \cdot 0 & - 9 + 9 \cdot 1 & - 1 + 9 (\frac{2}{15}) & 9 + 9 (- \frac{41}{15}) \end{array}]$

Schritt 3.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ 0 & 1 & \frac{2}{15} & - \frac{41}{15} \\ 0 & 0 & \frac{1}{5} & - \frac{78}{5} \end{array}]$

Schritt 3.5

Multiply each element of $R_{3}$ by $5$ to make the entry at $3, 3$ a $1$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.5.1

Multiply each element of $R_{3}$ by $5$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ 0 & 1 & \frac{2}{15} & - \frac{41}{15} \\ 5 \cdot 0 & 5 \cdot 0 & 5 (\frac{1}{5}) & 5 (- \frac{78}{5}) \end{array}]$

Schritt 3.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ 0 & 1 & \frac{2}{15} & - \frac{41}{15} \\ 0 & 0 & 1 & - 78 \end{array}]$

Schritt 3.6

Perform the row operation $R_{2} = R_{2} - \frac{2}{15} R_{3}$ to make the entry at $2, 3$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.6.1

Perform the row operation $R_{2} = R_{2} - \frac{2}{15} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ 0 - \frac{2}{15} \cdot 0 & 1 - \frac{2}{15} \cdot 0 & \frac{2}{15} - \frac{2}{15} \cdot 1 & - \frac{41}{15} - \frac{2}{15} \cdot - 78 \\ 0 & 0 & 1 & - 78 \end{array}]$

Schritt 3.6.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{20} & \frac{1}{20} & \frac{3}{10} \\ 0 & 1 & 0 & \frac{23}{3} \\ 0 & 0 & 1 & - 78 \end{array}]$

Schritt 3.7

Perform the row operation $R_{1} = R_{1} - \frac{1}{20} R_{3}$ to make the entry at $1, 3$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.7.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{20} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{20} \cdot 0 & \frac{9}{20} - \frac{1}{20} \cdot 0 & \frac{1}{20} - \frac{1}{20} \cdot 1 & \frac{3}{10} - \frac{1}{20} \cdot - 78 \\ 0 & 1 & 0 & \frac{23}{3} \\ 0 & 0 & 1 & - 78 \end{array}]$

Schritt 3.7.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{9}{20} & 0 & \frac{21}{5} \\ 0 & 1 & 0 & \frac{23}{3} \\ 0 & 0 & 1 & - 78 \end{array}]$

Schritt 3.8

Perform the row operation $R_{1} = R_{1} - \frac{9}{20} R_{2}$ to make the entry at $1, 2$ a $0$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 3.8.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{20} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{20} \cdot 0 & \frac{9}{20} - \frac{9}{20} \cdot 1 & 0 - \frac{9}{20} \cdot 0 & \frac{21}{5} - \frac{9}{20} \cdot \frac{23}{3} \\ 0 & 1 & 0 & \frac{23}{3} \\ 0 & 0 & 1 & - 78 \end{array}]$

Schritt 3.8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{3}{4} \\ 0 & 1 & 0 & \frac{23}{3} \\ 0 & 0 & 1 & - 78 \end{array}]$

Schritt 4

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

$x = \frac{3}{4}$

$y = \frac{23}{3}$

$z = - 78$

Schritt 5

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

$(\frac{3}{4}, \frac{23}{3}, - 78)$