Matematica discreta Esempi

$0.03 x + 0.04 y + 0.09 z = 647$ , $x + y + z = 10000$ , $3 x + 4 y + 9 z = 64700$

Passaggio 1

Write the system as a matrix.

$[\begin{array}{c} 0.03 & 0.04 & 0.09 & 647 \\ 1 & 1 & 1 & 10000 \\ 3 & 4 & 9 & 64700 \end{array}]$

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 $\frac{1}{0.03}$ to make the entry at $1, 1$ a $1$ .

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Passaggio 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{0.03}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{0.03}{0.03} & \frac{0.04}{0.03} & \frac{0.09}{0.03} & \frac{647}{0.03} \\ 1 & 1 & 1 & 10000 \\ 3 & 4 & 9 & 64700 \end{array}]$

Passaggio 2.1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 1 & 1 & 1 & 10000 \\ 3 & 4 & 9 & 64700 \end{array}]$

Passaggio 2.2

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

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Passaggio 2.2.1

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

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 1 - 1 & 1 - 1.\overline{3} & 1 - 3 & 10000 - 21566.\overline{6} \\ 3 & 4 & 9 & 64700 \end{array}]$

Passaggio 2.2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & - 0.\overline{3} & - 2 & - 11566.\overline{6} \\ 3 & 4 & 9 & 64700 \end{array}]$

Passaggio 2.3

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

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Passaggio 2.3.1

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

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & - 0.\overline{3} & - 2 & - 11566.\overline{6} \\ 3 - 3 \cdot 1 & 4 - 3 \cdot 1.\overline{3} & 9 - 3 \cdot 3 & 64700 - 3 \cdot 21566.\overline{6} \end{array}]$

Passaggio 2.3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & - 0.\overline{3} & - 2 & - 11566.\overline{6} \\ 0 & 0 & 0 & 0 \end{array}]$

Passaggio 2.4

Multiply each element of $R_{2}$ by $\frac{1}{- 0.\overline{3}}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{- 0.\overline{3}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ \frac{0}{- 0.\overline{3}} & \frac{- 0.\overline{3}}{- 0.\overline{3}} & \frac{- 2}{- 0.\overline{3}} & \frac{- 11566.\overline{6}}{- 0.\overline{3}} \\ 0 & 0 & 0 & 0 \end{array}]$

Passaggio 2.4.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & 1 & 6 & 34700 \\ 0 & 0 & 0 & 0 \end{array}]$

Passaggio 2.5

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

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Passaggio 2.5.1

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

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & 1 & 6 & 34700 \\ 00 \cdot 0 & 00 \cdot 1 & 00 \cdot 6 & 00 \cdot 34700 \end{array}]$

Passaggio 2.5.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & 1 & 6 & 34700 \\ 0 & 0 & 0 & 0 \end{array}]$

Passaggio 2.6

Multiply each element of $R_{3}$ by $\frac{1}{- 6 \cdot 10^{- 28}}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 2.6.1

Multiply each element of $R_{3}$ by $\frac{1}{- 6 \cdot 10^{- 28}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & 1 & 6 & 34700 \\ \frac{0}{- 6 \cdot 10^{- 28}} & \frac{0}{- 6 \cdot 10^{- 28}} & \frac{0}{- 6 \cdot 10^{- 28}} & \frac{0}{- 6 \cdot 10^{- 28}} \end{array}]$

Passaggio 2.6.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & 1 & 6 & 34700 \\ 0 & 0 & 1 & 7.45 \cdot 10^{3} \end{array}]$

Passaggio 2.7

Perform the row operation $R_{2} = R_{2} - 6 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Passaggio 2.7.1

Perform the row operation $R_{2} = R_{2} - 6 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 - 6 \cdot 0 & 1 - 6 \cdot 0 & 6 - 6 \cdot 1 & 34700 - 6 \cdot 7.45 \cdot 10^{3} \\ 0 & 0 & 1 & 7.45 \cdot 10^{3} \end{array}]$

Passaggio 2.7.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 3 & 21566.\overline{6} \\ 0 & 1 & 0 & - 1 \cdot 10^{4} \\ 0 & 0 & 1 & 7.45 \cdot 10^{3} \end{array}]$

Passaggio 2.8

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

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Passaggio 2.8.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 1.\overline{3} - 3 \cdot 0 & 3 - 3 \cdot 1 & 21566.\overline{6} - 3 \cdot 7.45 \cdot 10^{3} \\ 0 & 1 & 0 & - 1 \cdot 10^{4} \\ 0 & 0 & 1 & 7.45 \cdot 10^{3} \end{array}]$

Passaggio 2.8.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 1.\overline{3} & 0 & - 7.8\overline{3} \cdot 10^{2} \\ 0 & 1 & 0 & - 1 \cdot 10^{4} \\ 0 & 0 & 1 & 7.45 \cdot 10^{3} \end{array}]$

Passaggio 2.9

Perform the row operation $R_{1} = R_{1} - 1.\overline{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Passaggio 2.9.1

Perform the row operation $R_{1} = R_{1} - 1.\overline{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 1.\overline{3} \cdot 0 & 1.\overline{3} - 1.\overline{3} \cdot 1 & 0 - 1.\overline{3} \cdot 0 & - 7.8\overline{3} \cdot 10^{2} - 1.\overline{3} \cdot - 1 \cdot 10^{4} \\ 0 & 1 & 0 & - 1 \cdot 10^{4} \\ 0 & 0 & 1 & 7.45 \cdot 10^{3} \end{array}]$

Passaggio 2.9.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1.255 \cdot 10^{4} \\ 0 & 1 & 0 & - 1 \cdot 10^{4} \\ 0 & 0 & 1 & 7.45 \cdot 10^{3} \end{array}]$

Passaggio 3

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

$x = 1.255 \cdot 10^{4}$

$y = - 1 \cdot 10^{4}$

$z = 7.45 \cdot 10^{3}$

Passaggio 4

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

$(1.255 \cdot 10^{4}, - 1 \cdot 10^{4}, 7.45 \cdot 10^{3})$