Matemática discreta Exemplos

$x + y - 7 z = 2$ , $x - y - 3 z = 0$ , $\frac{2}{3} x - \frac{10}{3} z = 5$

Etapa 1

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

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Etapa 1.1

Simplifique cada termo.

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Etapa 1.1.1

Combine $\frac{2}{3}$ e $x$ .

$x + y - 7 z = 2$

$x - y - 3 z = 0$

$\frac{2 x}{3} - \frac{10}{3} z = 5$

Etapa 1.1.2

Combine $z$ e $\frac{10}{3}$ .

$x + y - 7 z = 2$

$x - y - 3 z = 0$

$\frac{2 x}{3} - \frac{z \cdot 10}{3} = 5$

Etapa 1.1.3

Mova $10$ para a esquerda de $z$ .

$x + y - 7 z = 2$

$x - y - 3 z = 0$

$\frac{2 x}{3} - \frac{10 z}{3} = 5$

$x + y - 7 z = 2$

$x - y - 3 z = 0$

$\frac{2 x}{3} - \frac{10 z}{3} = 5$

Etapa 1.2

Reordene os termos.

$x + y - 7 z = 2$

$x - y - 3 z = 0$

$\frac{2}{3} x - (\frac{10}{3} z) = 5$

Etapa 1.3

Remova os parênteses.

$x + y - 7 z = 2$

$x - y - 3 z = 0$

$\frac{2}{3} x - \frac{10}{3} z = 5$

$x + y - 7 z = 2$

$x - y - 3 z = 0$

$\frac{2}{3} x - \frac{10}{3} z = 5$

Etapa 2

Write the system as a matrix.

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

Etapa 3

Encontre a forma escalonada reduzida por linhas.

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Etapa 3.1

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

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Etapa 3.1.1

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

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

Etapa 3.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & - 2 & 4 & - 2 \\ \frac{2}{3} & 0 & - \frac{10}{3} & 5 \end{array}]$

Etapa 3.2

Perform the row operation $R_{3} = R_{3} - \frac{2}{3} R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 3.2.1

Perform the row operation $R_{3} = R_{3} - \frac{2}{3} R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & - 2 & 4 & - 2 \\ \frac{2}{3} - \frac{2}{3} \cdot 1 & 0 - \frac{2}{3} \cdot 1 & - \frac{10}{3} - \frac{2}{3} \cdot - 7 & 5 - \frac{2}{3} \cdot 2 \end{array}]$

Etapa 3.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & - 2 & 4 & - 2 \\ 0 & - \frac{2}{3} & \frac{4}{3} & \frac{11}{3} \end{array}]$

Etapa 3.3

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

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Etapa 3.3.1

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

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot - 2 \\ 0 & - \frac{2}{3} & \frac{4}{3} & \frac{11}{3} \end{array}]$

Etapa 3.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & 1 & - 2 & 1 \\ 0 & - \frac{2}{3} & \frac{4}{3} & \frac{11}{3} \end{array}]$

Etapa 3.4

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

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Etapa 3.4.1

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

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & 1 & - 2 & 1 \\ 0 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & \frac{4}{3} + \frac{2}{3} \cdot - 2 & \frac{11}{3} + \frac{2}{3} \cdot 1 \end{array}]$

Etapa 3.4.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & 1 & - 2 & 1 \\ 0 & 0 & 0 & \frac{13}{3} \end{array}]$

Etapa 3.5

Multiply each element of $R_{3}$ by $\frac{3}{13}$ to make the entry at $3, 4$ a $1$ .

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Etapa 3.5.1

Multiply each element of $R_{3}$ by $\frac{3}{13}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & 1 & - 2 & 1 \\ \frac{3}{13} \cdot 0 & \frac{3}{13} \cdot 0 & \frac{3}{13} \cdot 0 & \frac{3}{13} \cdot \frac{13}{3} \end{array}]$

Etapa 3.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & 1 & - 2 & 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 3.6

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

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Etapa 3.6.1

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

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 - 0 & 1 - 0 & - 2 - 0 & 1 - 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 3.6.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 7 & 2 \\ 0 & 1 & - 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 3.7

Perform the row operation $R_{1} = R_{1} - 2 R_{3}$ to make the entry at $1, 4$ a $0$ .

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Etapa 3.7.1

Perform the row operation $R_{1} = R_{1} - 2 R_{3}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 2 \cdot 0 & 1 - 2 \cdot 0 & - 7 - 2 \cdot 0 & 2 - 2 \cdot 1 \\ 0 & 1 & - 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 3.7.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 1 & - 7 & 0 \\ 0 & 1 & - 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 3.8

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

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Etapa 3.8.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & - 7 + 2 & 0 - 0 \\ 0 & 1 & - 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Etapa 3.8.2

Simplifique $R_{1}$ .

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

Etapa 4

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

$x - 5 z = 0$

$y - 2 z = 0$

$0 = 1$

Etapa 5

The system is inconsistent so there is no solution.

$Nenhuma solução$