Matemática discreta Exemplos

$- 2 x - y + 4 z = - 6$ , $5 y + 7 z = 20$ , $- 4 x + 6 y + 3 z = 20$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} - 2 & - 1 & 4 & - 6 \\ 0 & 5 & 7 & 20 \\ - 4 & 6 & 3 & 20 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

$[\begin{array}{c} - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 1 & - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot - 6 \\ 0 & 5 & 7 & 20 \\ - 4 & 6 & 3 & 20 \end{array}]$

Etapa 2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 2 & 3 \\ 0 & 5 & 7 & 20 \\ - 4 & 6 & 3 & 20 \end{array}]$

Etapa 2.2

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

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Etapa 2.2.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & - 2 & 3 \\ 0 & 5 & 7 & 20 \\ - 4 + 4 \cdot 1 & 6 + 4 (\frac{1}{2}) & 3 + 4 \cdot - 2 & 20 + 4 \cdot 3 \end{array}]$

Etapa 2.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 2 & 3 \\ 0 & 5 & 7 & 20 \\ 0 & 8 & - 5 & 32 \end{array}]$

Etapa 2.3

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 2 & 3 \\ \frac{0}{5} & \frac{5}{5} & \frac{7}{5} & \frac{20}{5} \\ 0 & 8 & - 5 & 32 \end{array}]$

Etapa 2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 2 & 3 \\ 0 & 1 & \frac{7}{5} & 4 \\ 0 & 8 & - 5 & 32 \end{array}]$

Etapa 2.4

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 2.4.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & - 2 & 3 \\ 0 & 1 & \frac{7}{5} & 4 \\ 0 - 8 \cdot 0 & 8 - 8 \cdot 1 & - 5 - 8 (\frac{7}{5}) & 32 - 8 \cdot 4 \end{array}]$

Etapa 2.4.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 2 & 3 \\ 0 & 1 & \frac{7}{5} & 4 \\ 0 & 0 & - \frac{81}{5} & 0 \end{array}]$

Etapa 2.5

Multiply each element of $R_{3}$ by $- \frac{5}{81}$ to make the entry at $3, 3$ a $1$ .

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Etapa 2.5.1

Multiply each element of $R_{3}$ by $- \frac{5}{81}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 2 & 3 \\ 0 & 1 & \frac{7}{5} & 4 \\ - \frac{5}{81} \cdot 0 & - \frac{5}{81} \cdot 0 & - \frac{5}{81} (- \frac{81}{5}) & - \frac{5}{81} \cdot 0 \end{array}]$

Etapa 2.5.2

Simplifique $R_{3}$ .

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

Etapa 2.6

Perform the row operation $R_{2} = R_{2} - \frac{7}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 2.6.1

Perform the row operation $R_{2} = R_{2} - \frac{7}{5} R_{3}$ to make the entry at $2, 3$ a $0$ .

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

Etapa 2.6.2

Simplifique $R_{2}$ .

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

Etapa 2.7

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

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Etapa 2.7.1

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

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

Etapa 2.7.2

Simplifique $R_{1}$ .

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

Etapa 2.8

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

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Etapa 2.8.1

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

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

Etapa 2.8.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$x = 1$

$y = 4$

$z = 0$

Etapa 4

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

$(1, 4, 0)$