Finite Mathematik Beispiele

$25 x + 100 y + 275 z + 100 w = 3675$ , $30 x + 80 y + 210 z + 80 w = 4710$ , $30 x + 80 y + 230 z + 60 w = 4110$

Schritt 1

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

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Schritt 1.1

Bewege $275 z$ .

$25 x + 100 y + 100 w + 275 z = 3675$

$30 x + 80 y + 210 z + 80 w = 4710$

$30 x + 80 y + 230 z + 60 w = 4110$

Schritt 1.2

Bewege $100 y$ .

$25 x + 100 w + 100 y + 275 z = 3675$

$30 x + 80 y + 210 z + 80 w = 4710$

$30 x + 80 y + 230 z + 60 w = 4110$

Schritt 1.3

Stelle $25 x$ und $100 w$ um.

$100 w + 25 x + 100 y + 275 z = 3675$

$30 x + 80 y + 210 z + 80 w = 4710$

$30 x + 80 y + 230 z + 60 w = 4110$

Schritt 1.4

Bewege $210 z$ .

$100 w + 25 x + 100 y + 275 z = 3675$

$30 x + 80 y + 80 w + 210 z = 4710$

$30 x + 80 y + 230 z + 60 w = 4110$

Schritt 1.5

Bewege $80 y$ .

$100 w + 25 x + 100 y + 275 z = 3675$

$30 x + 80 w + 80 y + 210 z = 4710$

$30 x + 80 y + 230 z + 60 w = 4110$

Schritt 1.6

Stelle $30 x$ und $80 w$ um.

$100 w + 25 x + 100 y + 275 z = 3675$

$80 w + 30 x + 80 y + 210 z = 4710$

$30 x + 80 y + 230 z + 60 w = 4110$

Schritt 1.7

Bewege $230 z$ .

$100 w + 25 x + 100 y + 275 z = 3675$

$80 w + 30 x + 80 y + 210 z = 4710$

$30 x + 80 y + 60 w + 230 z = 4110$

Schritt 1.8

Bewege $80 y$ .

$100 w + 25 x + 100 y + 275 z = 3675$

$80 w + 30 x + 80 y + 210 z = 4710$

$30 x + 60 w + 80 y + 230 z = 4110$

Schritt 1.9

Stelle $30 x$ und $60 w$ um.

$100 w + 25 x + 100 y + 275 z = 3675$

$80 w + 30 x + 80 y + 210 z = 4710$

$60 w + 30 x + 80 y + 230 z = 4110$

$100 w + 25 x + 100 y + 275 z = 3675$

$80 w + 30 x + 80 y + 210 z = 4710$

$60 w + 30 x + 80 y + 230 z = 4110$

Schritt 2

Write the system as a matrix.

$[\begin{array}{c} 100 & 25 & 100 & 275 & 3675 \\ 80 & 30 & 80 & 210 & 4710 \\ 60 & 30 & 80 & 230 & 4110 \end{array}]$

Schritt 3

Ermittele die normierte Zeilenstufenform.

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Schritt 3.1

Multiply each element of $R_{1}$ by $\frac{1}{100}$ to make the entry at $1, 1$ a $1$ .

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Schritt 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{100}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{100}{100} & \frac{25}{100} & \frac{100}{100} & \frac{275}{100} & \frac{3675}{100} \\ 80 & 30 & 80 & 210 & 4710 \\ 60 & 30 & 80 & 230 & 4110 \end{array}]$

Schritt 3.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 80 & 30 & 80 & 210 & 4710 \\ 60 & 30 & 80 & 230 & 4110 \end{array}]$

Schritt 3.2

Perform the row operation $R_{2} = R_{2} - 80 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 3.2.1

Perform the row operation $R_{2} = R_{2} - 80 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 80 - 80 \cdot 1 & 30 - 80 (\frac{1}{4}) & 80 - 80 \cdot 1 & 210 - 80 (\frac{11}{4}) & 4710 - 80 (\frac{147}{4}) \\ 60 & 30 & 80 & 230 & 4110 \end{array}]$

Schritt 3.2.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 0 & 10 & 0 & - 10 & 1770 \\ 60 & 30 & 80 & 230 & 4110 \end{array}]$

Schritt 3.3

Perform the row operation $R_{3} = R_{3} - 60 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 3.3.1

Perform the row operation $R_{3} = R_{3} - 60 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 0 & 10 & 0 & - 10 & 1770 \\ 60 - 60 \cdot 1 & 30 - 60 (\frac{1}{4}) & 80 - 60 \cdot 1 & 230 - 60 (\frac{11}{4}) & 4110 - 60 (\frac{147}{4}) \end{array}]$

Schritt 3.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 0 & 10 & 0 & - 10 & 1770 \\ 0 & 15 & 20 & 65 & 1905 \end{array}]$

Schritt 3.4

Multiply each element of $R_{2}$ by $\frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

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Schritt 3.4.1

Multiply each element of $R_{2}$ by $\frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ \frac{0}{10} & \frac{10}{10} & \frac{0}{10} & \frac{- 10}{10} & \frac{1770}{10} \\ 0 & 15 & 20 & 65 & 1905 \end{array}]$

Schritt 3.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 0 & 1 & 0 & - 1 & 177 \\ 0 & 15 & 20 & 65 & 1905 \end{array}]$

Schritt 3.5

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

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Schritt 3.5.1

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

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 0 & 1 & 0 & - 1 & 177 \\ 0 - 15 \cdot 0 & 15 - 15 \cdot 1 & 20 - 15 \cdot 0 & 65 - 15 \cdot - 1 & 1905 - 15 \cdot 177 \end{array}]$

Schritt 3.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 0 & 1 & 0 & - 1 & 177 \\ 0 & 0 & 20 & 80 & - 750 \end{array}]$

Schritt 3.6

Multiply each element of $R_{3}$ by $\frac{1}{20}$ to make the entry at $3, 3$ a $1$ .

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Schritt 3.6.1

Multiply each element of $R_{3}$ by $\frac{1}{20}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 0 & 1 & 0 & - 1 & 177 \\ \frac{0}{20} & \frac{0}{20} & \frac{20}{20} & \frac{80}{20} & \frac{- 750}{20} \end{array}]$

Schritt 3.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 1 & \frac{11}{4} & \frac{147}{4} \\ 0 & 1 & 0 & - 1 & 177 \\ 0 & 0 & 1 & 4 & - \frac{75}{2} \end{array}]$

Schritt 3.7

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

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Schritt 3.7.1

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

$[\begin{array}{c} 1 - 0 & \frac{1}{4} - 0 & 1 - 1 & \frac{11}{4} - 4 & \frac{147}{4} + \frac{75}{2} \\ 0 & 1 & 0 & - 1 & 177 \\ 0 & 0 & 1 & 4 & - \frac{75}{2} \end{array}]$

Schritt 3.7.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{4} & 0 & - \frac{5}{4} & \frac{297}{4} \\ 0 & 1 & 0 & - 1 & 177 \\ 0 & 0 & 1 & 4 & - \frac{75}{2} \end{array}]$

Schritt 3.8

Perform the row operation $R_{1} = R_{1} - \frac{1}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 3.8.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{4} \cdot 0 & \frac{1}{4} - \frac{1}{4} \cdot 1 & 0 - \frac{1}{4} \cdot 0 & - \frac{5}{4} - \frac{1}{4} \cdot - 1 & \frac{297}{4} - \frac{1}{4} \cdot 177 \\ 0 & 1 & 0 & - 1 & 177 \\ 0 & 0 & 1 & 4 & - \frac{75}{2} \end{array}]$

Schritt 3.8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 1 & 30 \\ 0 & 1 & 0 & - 1 & 177 \\ 0 & 0 & 1 & 4 & - \frac{75}{2} \end{array}]$

Schritt 4

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

$w - z = 30$

$x - z = 177$

$y + 4 z = - \frac{75}{2}$

Schritt 5

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

$(30 + z, 177 + z, - \frac{75}{2} - 4 z, z)$