Matematica discreta Esempi

$0.5 x + 0.1 y = 0.9$ , $0.1 x - 0.1 y = - 0.1$ , $x + y = \frac{11}{3}$

Passaggio 1

Write the system as a matrix.

$[\begin{array}{c} 0.5 & 0.1 & 0.9 \\ 0.1 & - 0.1 & - 0.1 \\ 1 & 1 & \frac{11}{3} \end{array}]$

Passaggio 2

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 2.1

Multiply each element of $R_{1}$ by $\frac{1}{0.5}$ to make the entry at $1, 1$ a $1$ .

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Passaggio 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{0.5}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{0.5}{0.5} & \frac{0.1}{0.5} & \frac{0.9}{0.5} \\ 0.1 & - 0.1 & - 0.1 \\ 1 & 1 & \frac{11}{3} \end{array}]$

Passaggio 2.1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0.1 & - 0.1 & - 0.1 \\ 1 & 1 & \frac{11}{3} \end{array}]$

Passaggio 2.2

Perform the row operation $R_{2} = R_{2} - 0.1 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Passaggio 2.2.1

Perform the row operation $R_{2} = R_{2} - 0.1 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0.1 - 0.1 \cdot 1 & - 0.1 - 0.1 \cdot 0.2 & - 0.1 - 0.1 \cdot 1.8 \\ 1 & 1 & \frac{11}{3} \end{array}]$

Passaggio 2.2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & - 0.12 & - 0.28 \\ 1 & 1 & \frac{11}{3} \end{array}]$

Passaggio 2.3

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

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Passaggio 2.3.1

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

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & - 0.12 & - 0.28 \\ 1 - 1 & 1 - 0.2 & \frac{11}{3} - 1.8 \end{array}]$

Passaggio 2.3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & - 0.12 & - 0.28 \\ 0 & 0.8 & 1.8\overline{6} \end{array}]$

Passaggio 2.4

Multiply each element of $R_{2}$ by $\frac{1}{- 0.12}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{- 0.12}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ \frac{0}{- 0.12} & \frac{- 0.12}{- 0.12} & \frac{- 0.28}{- 0.12} \\ 0 & 0.8 & 1.8\overline{6} \end{array}]$

Passaggio 2.4.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & 1 & 2.\overline{3} \\ 0 & 0.8 & 1.8\overline{6} \end{array}]$

Passaggio 2.5

Perform the row operation $R_{3} = R_{3} - 0.8 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Passaggio 2.5.1

Perform the row operation $R_{3} = R_{3} - 0.8 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & 1 & 2.\overline{3} \\ 0 - 0.8 \cdot 0 & 0.8 - 0.8 \cdot 1 & 1.8\overline{6} - 0.8 \cdot 2.\overline{3} \end{array}]$

Passaggio 2.5.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & 1 & 2.\overline{3} \\ 0 & 0 & 0 \end{array}]$

Passaggio 2.6

Multiply each element of $R_{3}$ by $\frac{1}{1 \cdot 10^{- 28}}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 2.6.1

Multiply each element of $R_{3}$ by $\frac{1}{1 \cdot 10^{- 28}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & 1 & 2.\overline{3} \\ \frac{0}{1 \cdot 10^{- 28}} & \frac{0}{1 \cdot 10^{- 28}} & \frac{0}{1 \cdot 10^{- 28}} \end{array}]$

Passaggio 2.6.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & 1 & 2.\overline{3} \\ 0 & 0 & 1 \end{array}]$

Passaggio 2.7

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

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Passaggio 2.7.1

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

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 - 2.\overline{3} \cdot 0 & 1 - 2.\overline{3} \cdot 0 & 2.\overline{3} - 2.\overline{3} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Passaggio 2.7.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 0.2 & 1.8 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Passaggio 2.8

Perform the row operation $R_{1} = R_{1} - 1.8 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Passaggio 2.8.1

Perform the row operation $R_{1} = R_{1} - 1.8 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 1.8 \cdot 0 & 0.2 - 1.8 \cdot 0 & 1.8 - 1.8 \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Passaggio 2.8.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0.2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Passaggio 2.9

Perform the row operation $R_{1} = R_{1} - 0.2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Passaggio 2.9.1

Perform the row operation $R_{1} = R_{1} - 0.2 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0.2 \cdot 0 & 0.2 - 0.2 \cdot 1 & 0 - 0.2 \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Passaggio 2.9.2

Semplifica $R_{1}$ .

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

Passaggio 3

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

$x = 0$

$y = 0$

$0 = 1$

Passaggio 4

The system is inconsistent so there is no solution.

$Nessuna soluzione$