Matematica discreta Esempi

$x - 2 y + 3 z = 21$ , $y - z + w = - 17$ , $- 3 x + 3 y - 3 z + 4 w = - 28$ , $2 y - 3 z + w = - 22$

Passaggio 1

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

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Passaggio 1.1

Sposta $- z$ .

$x - 2 y + 3 z = 21$

$y + w - z = - 17$

$- 3 x + 3 y - 3 z + 4 w = - 28$

$2 y - 3 z + w = - 22$

Passaggio 1.2

Riordina $y$ e $w$ .

$x - 2 y + 3 z = 21$

$w + y - z = - 17$

$- 3 x + 3 y - 3 z + 4 w = - 28$

$2 y - 3 z + w = - 22$

Passaggio 1.3

Sposta $- 3 z$ .

$x - 2 y + 3 z = 21$

$w + y - z = - 17$

$- 3 x + 3 y + 4 w - 3 z = - 28$

$2 y - 3 z + w = - 22$

Passaggio 1.4

Sposta $3 y$ .

$x - 2 y + 3 z = 21$

$w + y - z = - 17$

$- 3 x + 4 w + 3 y - 3 z = - 28$

$2 y - 3 z + w = - 22$

Passaggio 1.5

Riordina $- 3 x$ e $4 w$ .

$x - 2 y + 3 z = 21$

$w + y - z = - 17$

$4 w - 3 x + 3 y - 3 z = - 28$

$2 y - 3 z + w = - 22$

Passaggio 1.6

Sposta $- 3 z$ .

$x - 2 y + 3 z = 21$

$w + y - z = - 17$

$4 w - 3 x + 3 y - 3 z = - 28$

$2 y + w - 3 z = - 22$

Passaggio 1.7

Riordina $2 y$ e $w$ .

$x - 2 y + 3 z = 21$

$w + y - z = - 17$

$4 w - 3 x + 3 y - 3 z = - 28$

$w + 2 y - 3 z = - 22$

$x - 2 y + 3 z = 21$

$w + y - z = - 17$

$4 w - 3 x + 3 y - 3 z = - 28$

$w + 2 y - 3 z = - 22$

Passaggio 2

Write the system as a matrix.

$[\begin{array}{c} 0 & 1 & - 2 & 3 & 21 \\ 1 & 0 & 1 & - 1 & - 17 \\ 4 & - 3 & 3 & - 3 & - 28 \\ 1 & 0 & 2 & - 3 & - 22 \end{array}]$

Passaggio 3

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 3.1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 4 & - 3 & 3 & - 3 & - 28 \\ 1 & 0 & 2 & - 3 & - 22 \end{array}]$

Passaggio 3.2

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Passaggio 3.2.1

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 4 - 4 \cdot 1 & - 3 - 4 \cdot 0 & 3 - 4 \cdot 1 & - 3 - 4 \cdot - 1 & - 28 - 4 \cdot - 17 \\ 1 & 0 & 2 & - 3 & - 22 \end{array}]$

Passaggio 3.2.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & - 3 & - 1 & 1 & 40 \\ 1 & 0 & 2 & - 3 & - 22 \end{array}]$

Passaggio 3.3

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

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Passaggio 3.3.1

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

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & - 3 & - 1 & 1 & 40 \\ 1 - 1 & 0 - 0 & 2 - 1 & - 3 + 1 & - 22 + 17 \end{array}]$

Passaggio 3.3.2

Semplifica $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & - 3 & - 1 & 1 & 40 \\ 0 & 0 & 1 & - 2 & - 5 \end{array}]$

Passaggio 3.4

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

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Passaggio 3.4.1

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

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 1 + 3 \cdot - 2 & 1 + 3 \cdot 3 & 40 + 3 \cdot 21 \\ 0 & 0 & 1 & - 2 & - 5 \end{array}]$

Passaggio 3.4.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & 0 & - 7 & 10 & 103 \\ 0 & 0 & 1 & - 2 & - 5 \end{array}]$

Passaggio 3.5

Multiply each element of $R_{3}$ by $- \frac{1}{7}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 3.5.1

Multiply each element of $R_{3}$ by $- \frac{1}{7}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ - \frac{1}{7} \cdot 0 & - \frac{1}{7} \cdot 0 & - \frac{1}{7} \cdot - 7 & - \frac{1}{7} \cdot 10 & - \frac{1}{7} \cdot 103 \\ 0 & 0 & 1 & - 2 & - 5 \end{array}]$

Passaggio 3.5.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{103}{7} \\ 0 & 0 & 1 & - 2 & - 5 \end{array}]$

Passaggio 3.6

Perform the row operation $R_{4} = R_{4} - R_{3}$ to make the entry at $4, 3$ a $0$ .

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Passaggio 3.6.1

Perform the row operation $R_{4} = R_{4} - R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{103}{7} \\ 0 - 0 & 0 - 0 & 1 - 1 & - 2 + \frac{10}{7} & - 5 + \frac{103}{7} \end{array}]$

Passaggio 3.6.2

Semplifica $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{103}{7} \\ 0 & 0 & 0 & - \frac{4}{7} & \frac{68}{7} \end{array}]$

Passaggio 3.7

Multiply each element of $R_{4}$ by $- \frac{7}{4}$ to make the entry at $4, 4$ a $1$ .

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Passaggio 3.7.1

Multiply each element of $R_{4}$ by $- \frac{7}{4}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{103}{7} \\ - \frac{7}{4} \cdot 0 & - \frac{7}{4} \cdot 0 & - \frac{7}{4} \cdot 0 & - \frac{7}{4} (- \frac{4}{7}) & - \frac{7}{4} \cdot \frac{68}{7} \end{array}]$

Passaggio 3.7.2

Semplifica $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & 0 & 1 & - \frac{10}{7} & - \frac{103}{7} \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.8

Perform the row operation $R_{3} = R_{3} + \frac{10}{7} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Passaggio 3.8.1

Perform the row operation $R_{3} = R_{3} + \frac{10}{7} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 + \frac{10}{7} \cdot 0 & 0 + \frac{10}{7} \cdot 0 & 1 + \frac{10}{7} \cdot 0 & - \frac{10}{7} + \frac{10}{7} \cdot 1 & - \frac{103}{7} + \frac{10}{7} \cdot - 17 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.8.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 3 & 21 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.9

Perform the row operation $R_{2} = R_{2} - 3 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Passaggio 3.9.1

Perform the row operation $R_{2} = R_{2} - 3 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 - 3 \cdot 0 & 1 - 3 \cdot 0 & - 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & 21 - 3 \cdot - 17 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.9.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 & - 17 \\ 0 & 1 & - 2 & 0 & 72 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.10

Perform the row operation $R_{1} = R_{1} + R_{4}$ to make the entry at $1, 4$ a $0$ .

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Passaggio 3.10.1

Perform the row operation $R_{1} = R_{1} + R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + 0 & 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & - 17 - 17 \\ 0 & 1 & - 2 & 0 & 72 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.10.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 34 \\ 0 & 1 & - 2 & 0 & 72 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.11

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

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Passaggio 3.11.1

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

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 34 \\ 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & 72 + 2 \cdot - 39 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.11.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & 0 & - 34 \\ 0 & 1 & 0 & 0 & - 6 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.12

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

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Passaggio 3.12.1

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

$[\begin{array}{c} 1 - 0 & 0 - 0 & 1 - 1 & 0 - 0 & - 34 + 39 \\ 0 & 1 & 0 & 0 & - 6 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 3.12.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 5 \\ 0 & 1 & 0 & 0 & - 6 \\ 0 & 0 & 1 & 0 & - 39 \\ 0 & 0 & 0 & 1 & - 17 \end{array}]$

Passaggio 4

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

$w = 5$

$x = - 6$

$y = - 39$

$z = - 17$

Passaggio 5

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

$(5, - 6, - 39, - 17)$