Matemática discreta Exemplos

$x + y + z = 102$ , $2 x + 3 y + 4 z = 303$ , $11 x + 17 y + 23 z = 1716$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 1 & 102 \\ 2 & 3 & 4 & 303 \\ 11 & 17 & 23 & 1716 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

$[\begin{array}{c} 1 & 1 & 1 & 102 \\ 2 - 2 \cdot 1 & 3 - 2 \cdot 1 & 4 - 2 \cdot 1 & 303 - 2 \cdot 102 \\ 11 & 17 & 23 & 1716 \end{array}]$

Etapa 2.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 102 \\ 0 & 1 & 2 & 99 \\ 11 & 17 & 23 & 1716 \end{array}]$

Etapa 2.2

Perform the row operation $R_{3} = R_{3} - 11 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 2.2.1

Perform the row operation $R_{3} = R_{3} - 11 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 102 \\ 0 & 1 & 2 & 99 \\ 11 - 11 \cdot 1 & 17 - 11 \cdot 1 & 23 - 11 \cdot 1 & 1716 - 11 \cdot 102 \end{array}]$

Etapa 2.2.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 102 \\ 0 & 1 & 2 & 99 \\ 0 & 6 & 12 & 594 \end{array}]$

Etapa 2.3

Perform the row operation $R_{3} = R_{3} - 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 2.3.1

Perform the row operation $R_{3} = R_{3} - 6 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 102 \\ 0 & 1 & 2 & 99 \\ 0 - 6 \cdot 0 & 6 - 6 \cdot 1 & 12 - 6 \cdot 2 & 594 - 6 \cdot 99 \end{array}]$

Etapa 2.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 102 \\ 0 & 1 & 2 & 99 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 2.4

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

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Etapa 2.4.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 1 - 2 & 102 - 99 \\ 0 & 1 & 2 & 99 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 2.4.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 1 & 3 \\ 0 & 1 & 2 & 99 \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 3

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

$x - z = 3$

$y + 2 z = 99$

$0 = 0$

Etapa 4

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

$(3 + z, 99 - 2 z, z)$