Matemática discreta Exemplos

$x - y - 8 z = 0$ , $y + 4 z = 8$ , $3 y + 14 z = 22$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 1 & - 8 & 0 \\ 0 & 1 & 4 & 8 \\ 0 & 3 & 14 & 22 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

$[\begin{array}{c} 1 & - 1 & - 8 & 0 \\ 0 & 1 & 4 & 8 \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & 14 - 3 \cdot 4 & 22 - 3 \cdot 8 \end{array}]$

Etapa 2.1.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 8 & 0 \\ 0 & 1 & 4 & 8 \\ 0 & 0 & 2 & - 2 \end{array}]$

Etapa 2.2

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

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Etapa 2.2.1

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

$[\begin{array}{c} 1 & - 1 & - 8 & 0 \\ 0 & 1 & 4 & 8 \\ \frac{0}{2} & \frac{0}{2} & \frac{2}{2} & \frac{- 2}{2} \end{array}]$

Etapa 2.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 8 & 0 \\ 0 & 1 & 4 & 8 \\ 0 & 0 & 1 & - 1 \end{array}]$

Etapa 2.3

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

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Etapa 2.3.1

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

$[\begin{array}{c} 1 & - 1 & - 8 & 0 \\ 0 - 4 \cdot 0 & 1 - 4 \cdot 0 & 4 - 4 \cdot 1 & 8 - 4 \cdot - 1 \\ 0 & 0 & 1 & - 1 \end{array}]$

Etapa 2.3.2

Simplifique $R_{2}$ .

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

Etapa 2.4

Perform the row operation $R_{1} = R_{1} + 8 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Etapa 2.4.1

Perform the row operation $R_{1} = R_{1} + 8 R_{3}$ to make the entry at $1, 3$ a $0$ .

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

Etapa 2.4.2

Simplifique $R_{1}$ .

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

Etapa 2.5

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

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Etapa 2.5.1

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

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

Etapa 2.5.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 12 \\ 0 & 0 & 1 & - 1 \end{array}]$

Etapa 3

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

$x = 4$

$y = 12$

$z = - 1$

Etapa 4

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

$(4, 12, - 1)$