Matematica discreta Esempi

$0.4 x + 0.4 y + 0.2 z = 17800$ , $0.5 y + 0.5 z = 31000$ , $0.2 x + 0.3 y + 0.5 z = 39200$

Passaggio 1

Write the system as a matrix.

$[\begin{array}{c} 0.4 & 0.4 & 0.2 & 17800 \\ 0 & 0.5 & 0.5 & 31000 \\ 0.2 & 0.3 & 0.5 & 39200 \end{array}]$

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 $\frac{1}{0.4}$ to make the entry at $1, 1$ a $1$ .

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Passaggio 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{0.4}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{0.4}{0.4} & \frac{0.4}{0.4} & \frac{0.2}{0.4} & \frac{17800}{0.4} \\ 0 & 0.5 & 0.5 & 31000 \\ 0.2 & 0.3 & 0.5 & 39200 \end{array}]$

Passaggio 2.1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 0.5 & 0.5 & 31000 \\ 0.2 & 0.3 & 0.5 & 39200 \end{array}]$

Passaggio 2.2

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Passaggio 2.2.1

Perform the row operation $R_{3} = R_{3} - 0.2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 0.5 & 0.5 & 31000 \\ 0.2 - 0.2 \cdot 1 & 0.3 - 0.2 \cdot 1 & 0.5 - 0.2 \cdot 0.5 & 39200 - 0.2 \cdot 44500 \end{array}]$

Passaggio 2.2.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 0.5 & 0.5 & 31000 \\ 0 & 0.1 & 0.4 & 30300 \end{array}]$

Passaggio 2.3

Multiply each element of $R_{2}$ by $\frac{1}{0.5}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{0.5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ \frac{0}{0.5} & \frac{0.5}{0.5} & \frac{0.5}{0.5} & \frac{31000}{0.5} \\ 0 & 0.1 & 0.4 & 30300 \end{array}]$

Passaggio 2.3.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 1 & 1 & 62000 \\ 0 & 0.1 & 0.4 & 30300 \end{array}]$

Passaggio 2.4

Perform the row operation $R_{3} = R_{3} - 0.1 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Passaggio 2.4.1

Perform the row operation $R_{3} = R_{3} - 0.1 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 1 & 1 & 62000 \\ 0 - 0.1 \cdot 0 & 0.1 - 0.1 \cdot 1 & 0.4 - 0.1 \cdot 1 & 30300 - 0.1 \cdot 62000 \end{array}]$

Passaggio 2.4.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 1 & 1 & 62000 \\ 0 & 0 & 0.3 & 24100 \end{array}]$

Passaggio 2.5

Multiply each element of $R_{3}$ by $\frac{1}{0.3}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 2.5.1

Multiply each element of $R_{3}$ by $\frac{1}{0.3}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 1 & 1 & 62000 \\ \frac{0}{0.3} & \frac{0}{0.3} & \frac{0.3}{0.3} & \frac{24100}{0.3} \end{array}]$

Passaggio 2.5.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 1 & 1 & 62000 \\ 0 & 0 & 1 & 80333.\overline{3} \end{array}]$

Passaggio 2.6

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

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Passaggio 2.6.1

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

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 - 0 & 1 - 0 & 1 - 1 & 62000 - 80333.\overline{3} \\ 0 & 0 & 1 & 80333.\overline{3} \end{array}]$

Passaggio 2.6.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 0.5 & 44500 \\ 0 & 1 & 0 & - 18333.\overline{3} \\ 0 & 0 & 1 & 80333.\overline{3} \end{array}]$

Passaggio 2.7

Perform the row operation $R_{1} = R_{1} - 0.5 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Passaggio 2.7.1

Perform the row operation $R_{1} = R_{1} - 0.5 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0.5 \cdot 0 & 1 - 0.5 \cdot 0 & 0.5 - 0.5 \cdot 1 & 44500 - 0.5 \cdot 80333.\overline{3} \\ 0 & 1 & 0 & - 18333.\overline{3} \\ 0 & 0 & 1 & 80333.\overline{3} \end{array}]$

Passaggio 2.7.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 4333.\overline{3} \\ 0 & 1 & 0 & - 18333.\overline{3} \\ 0 & 0 & 1 & 80333.\overline{3} \end{array}]$

Passaggio 2.8

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

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Passaggio 2.8.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 4333.\overline{3} + 18333.\overline{3} \\ 0 & 1 & 0 & - 18333.\overline{3} \\ 0 & 0 & 1 & 80333.\overline{3} \end{array}]$

Passaggio 2.8.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 22666.\overline{6} \\ 0 & 1 & 0 & - 18333.\overline{3} \\ 0 & 0 & 1 & 80333.\overline{3} \end{array}]$

Passaggio 3

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

$x = 22666.\overline{6}$

$y = - 18333.\overline{3}$

$z = 80333.\overline{3}$

Passaggio 4

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

$(22666.\overline{6}, - 18333.\overline{3}, 80333.\overline{3})$