Matemática discreta Exemplos

$5 x - y - z = 0$ , $x + y + z = 6$

Etapa 1

Write the system as a matrix.

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

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

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

Etapa 2.1.2

Simplifique $R_{1}$ .

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

Etapa 2.2

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

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Etapa 2.2.1

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

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

Etapa 2.2.2

Simplifique $R_{2}$ .

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

Etapa 2.3

Multiply each element of $R_{2}$ by $\frac{5}{6}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.3.1

Multiply each element of $R_{2}$ by $\frac{5}{6}$ to make the entry at $2, 2$ a $1$ .

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

Etapa 2.3.2

Simplifique $R_{2}$ .

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

Etapa 2.4

Perform the row operation $R_{1} = R_{1} + \frac{1}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 2.4.1

Perform the row operation $R_{1} = R_{1} + \frac{1}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Etapa 2.4.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$x = 1$

$y + z = 5$

Etapa 4

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

$(1, 5 - z, z)$