Álgebra linear Exemplos

$x + 2 y + 3 z = 2$ , $2 x + y - 2 = 0$ , $5 x + 2 y + 2 z = 5$

Etapa 1

Some $2$ aos dois lados da equação.

$x + 2 y + 3 z = 2, 2 x + y = 2, 5 x + 2 y + 2 z = 5$

Etapa 2

Escreva o sistema de equações em formato de matriz.

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

Etapa 3

Encontre a forma escalonada reduzida por linhas.

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Etapa 3.1

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

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Etapa 3.1.1

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

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

Etapa 3.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & - 3 & - 6 & - 2 \\ 5 & 2 & 2 & 5 \end{array}]$

Etapa 3.2

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 3.2.1

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & - 3 & - 6 & - 2 \\ 5 - 5 \cdot 1 & 2 - 5 \cdot 2 & 2 - 5 \cdot 3 & 5 - 5 \cdot 2 \end{array}]$

Etapa 3.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & - 3 & - 6 & - 2 \\ 0 & - 8 & - 13 & - 5 \end{array}]$

Etapa 3.3

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

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Etapa 3.3.1

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

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 6 & - \frac{1}{3} \cdot - 2 \\ 0 & - 8 & - 13 & - 5 \end{array}]$

Etapa 3.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & 1 & 2 & \frac{2}{3} \\ 0 & - 8 & - 13 & - 5 \end{array}]$

Etapa 3.4

Perform the row operation $R_{3} = R_{3} + 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 3.4.1

Perform the row operation $R_{3} = R_{3} + 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & 1 & 2 & \frac{2}{3} \\ 0 + 8 \cdot 0 & - 8 + 8 \cdot 1 & - 13 + 8 \cdot 2 & - 5 + 8 (\frac{2}{3}) \end{array}]$

Etapa 3.4.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & 1 & 2 & \frac{2}{3} \\ 0 & 0 & 3 & \frac{1}{3} \end{array}]$

Etapa 3.5

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

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Etapa 3.5.1

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

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & 1 & 2 & \frac{2}{3} \\ \frac{0}{3} & \frac{0}{3} & \frac{3}{3} & \frac{\frac{1}{3}}{3} \end{array}]$

Etapa 3.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & 1 & 2 & \frac{2}{3} \\ 0 & 0 & 1 & \frac{1}{9} \end{array}]$

Etapa 3.6

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

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Etapa 3.6.1

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

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & \frac{2}{3} - 2 (\frac{1}{9}) \\ 0 & 0 & 1 & \frac{1}{9} \end{array}]$

Etapa 3.6.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 2 \\ 0 & 1 & 0 & \frac{4}{9} \\ 0 & 0 & 1 & \frac{1}{9} \end{array}]$

Etapa 3.7

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

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Etapa 3.7.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & 2 - 3 (\frac{1}{9}) \\ 0 & 1 & 0 & \frac{4}{9} \\ 0 & 0 & 1 & \frac{1}{9} \end{array}]$

Etapa 3.7.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & \frac{5}{3} \\ 0 & 1 & 0 & \frac{4}{9} \\ 0 & 0 & 1 & \frac{1}{9} \end{array}]$

Etapa 3.8

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

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Etapa 3.8.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & \frac{5}{3} - 2 (\frac{4}{9}) \\ 0 & 1 & 0 & \frac{4}{9} \\ 0 & 0 & 1 & \frac{1}{9} \end{array}]$

Etapa 3.8.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{7}{9} \\ 0 & 1 & 0 & \frac{4}{9} \\ 0 & 0 & 1 & \frac{1}{9} \end{array}]$

Etapa 4

Use a matriz de resultados para declarar as soluções finais ao sistema de equações.

$x = \frac{7}{9}$

$y = \frac{4}{9}$

$z = \frac{1}{9}$

Etapa 5

A solução é o conjunto de pares ordenados que tornam o sistema verdadeiro.

$(\frac{7}{9}, \frac{4}{9}, \frac{1}{9})$

Etapa 6

Para decompor um vetor da solução, reorganize cada equação representada na forma de linha reduzida da matriz aumentada resolvendo a variável dependente em cada linha que produz igualdade vetorial.

$X = [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} \frac{7}{9} \\ \frac{4}{9} \\ \frac{1}{9} \end{array}]$