Matemática discreta Exemplos

$x + 8 y + 8 z = 8$ , $7 x + 7 y + z = 1$ , $8 x + 15 y + 9 z = - 9$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 7 & 7 & 1 & 1 \\ 8 & 15 & 9 & - 9 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

Perform the row operation $R_{2} = R_{2} - 7 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 2.1.1

Perform the row operation $R_{2} = R_{2} - 7 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 7 - 7 \cdot 1 & 7 - 7 \cdot 8 & 1 - 7 \cdot 8 & 1 - 7 \cdot 8 \\ 8 & 15 & 9 & - 9 \end{array}]$

Etapa 2.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & - 49 & - 55 & - 55 \\ 8 & 15 & 9 & - 9 \end{array}]$

Etapa 2.2

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

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Etapa 2.2.1

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

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & - 49 & - 55 & - 55 \\ 8 - 8 \cdot 1 & 15 - 8 \cdot 8 & 9 - 8 \cdot 8 & - 9 - 8 \cdot 8 \end{array}]$

Etapa 2.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & - 49 & - 55 & - 55 \\ 0 & - 49 & - 55 & - 73 \end{array}]$

Etapa 2.3

Multiply each element of $R_{2}$ by $- \frac{1}{49}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{49}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ - \frac{1}{49} \cdot 0 & - \frac{1}{49} \cdot - 49 & - \frac{1}{49} \cdot - 55 & - \frac{1}{49} \cdot - 55 \\ 0 & - 49 & - 55 & - 73 \end{array}]$

Etapa 2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & 1 & \frac{55}{49} & \frac{55}{49} \\ 0 & - 49 & - 55 & - 73 \end{array}]$

Etapa 2.4

Perform the row operation $R_{3} = R_{3} + 49 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 2.4.1

Perform the row operation $R_{3} = R_{3} + 49 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & 1 & \frac{55}{49} & \frac{55}{49} \\ 0 + 49 \cdot 0 & - 49 + 49 \cdot 1 & - 55 + 49 (\frac{55}{49}) & - 73 + 49 (\frac{55}{49}) \end{array}]$

Etapa 2.4.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & 1 & \frac{55}{49} & \frac{55}{49} \\ 0 & 0 & 0 & - 18 \end{array}]$

Etapa 2.5

Multiply each element of $R_{3}$ by $- \frac{1}{18}$ to make the entry at $3, 4$ a $1$ .

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Etapa 2.5.1

Multiply each element of $R_{3}$ by $- \frac{1}{18}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & 1 & \frac{55}{49} & \frac{55}{49} \\ - \frac{1}{18} \cdot 0 & - \frac{1}{18} \cdot 0 & - \frac{1}{18} \cdot 0 & - \frac{1}{18} \cdot - 18 \end{array}]$

Etapa 2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & 1 & \frac{55}{49} & \frac{55}{49} \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 2.6

Perform the row operation $R_{2} = R_{2} - \frac{55}{49} R_{3}$ to make the entry at $2, 4$ a $0$ .

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Etapa 2.6.1

Perform the row operation $R_{2} = R_{2} - \frac{55}{49} R_{3}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 - \frac{55}{49} \cdot 0 & 1 - \frac{55}{49} \cdot 0 & \frac{55}{49} - \frac{55}{49} \cdot 0 & \frac{55}{49} - \frac{55}{49} \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 2.6.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 8 & 8 & 8 \\ 0 & 1 & \frac{55}{49} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 2.7

Perform the row operation $R_{1} = R_{1} - 8 R_{3}$ to make the entry at $1, 4$ a $0$ .

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Etapa 2.7.1

Perform the row operation $R_{1} = R_{1} - 8 R_{3}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 8 \cdot 0 & 8 - 8 \cdot 0 & 8 - 8 \cdot 0 & 8 - 8 \cdot 1 \\ 0 & 1 & \frac{55}{49} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 2.7.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 8 & 8 & 0 \\ 0 & 1 & \frac{55}{49} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 2.8

Perform the row operation $R_{1} = R_{1} - 8 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 2.8.1

Perform the row operation $R_{1} = R_{1} - 8 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 8 \cdot 0 & 8 - 8 \cdot 1 & 8 - 8 (\frac{55}{49}) & 0 - 8 \cdot 0 \\ 0 & 1 & \frac{55}{49} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 2.8.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{48}{49} & 0 \\ 0 & 1 & \frac{55}{49} & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 3

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

$x - \frac{48}{49} z = 0$

$y + \frac{55}{49} z = 0$

$0 = 1$

Etapa 4

The system is inconsistent so there is no solution.

$Nenhuma solução$