Matematica discreta Esempi

$x - y + 12 z = 8$ , $y - 14 z = - 13$ , $z = 2$

Passaggio 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 1 & 12 & 8 \\ 0 & 1 & - 14 & - 13 \\ 0 & 0 & 1 & 2 \end{array}]$

Passaggio 2

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 2.1

Perform the row operation $R_{2} = R_{2} + 14 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Passaggio 2.1.1

Perform the row operation $R_{2} = R_{2} + 14 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 12 & 8 \\ 0 + 14 \cdot 0 & 1 + 14 \cdot 0 & - 14 + 14 \cdot 1 & - 13 + 14 \cdot 2 \\ 0 & 0 & 1 & 2 \end{array}]$

Passaggio 2.1.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 12 & 8 \\ 0 & 1 & 0 & 15 \\ 0 & 0 & 1 & 2 \end{array}]$

Passaggio 2.2

Perform the row operation $R_{1} = R_{1} - 12 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Passaggio 2.2.1

Perform the row operation $R_{1} = R_{1} - 12 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 12 \cdot 0 & - 1 - 12 \cdot 0 & 12 - 12 \cdot 1 & 8 - 12 \cdot 2 \\ 0 & 1 & 0 & 15 \\ 0 & 0 & 1 & 2 \end{array}]$

Passaggio 2.2.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 16 \\ 0 & 1 & 0 & 15 \\ 0 & 0 & 1 & 2 \end{array}]$

Passaggio 2.3

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

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Passaggio 2.3.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & - 16 + 1 \cdot 15 \\ 0 & 1 & 0 & 15 \\ 0 & 0 & 1 & 2 \end{array}]$

Passaggio 2.3.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 1 \\ 0 & 1 & 0 & 15 \\ 0 & 0 & 1 & 2 \end{array}]$

Passaggio 3

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

$x = - 1$

$y = 15$

$z = 2$

Passaggio 4

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

$(- 1, 15, 2)$