Matemática discreta Exemplos

$6 m - n = 38$ , $m - 6 n = - 17$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 6 & - 1 & 38 \\ 1 & - 6 & - 17 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

$[\begin{array}{c} \frac{6}{6} & \frac{- 1}{6} & \frac{38}{6} \\ 1 & - 6 & - 17 \end{array}]$

Etapa 2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{6} & \frac{19}{3} \\ 1 & - 6 & - 17 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - \frac{1}{6} & \frac{19}{3} \\ 1 - 1 & - 6 + \frac{1}{6} & - 17 - \frac{19}{3} \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{6} & \frac{19}{3} \\ 0 & - \frac{35}{6} & - \frac{70}{3} \end{array}]$

Etapa 2.3

Multiply each element of $R_{2}$ by $- \frac{6}{35}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.3.1

Multiply each element of $R_{2}$ by $- \frac{6}{35}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{6} & \frac{19}{3} \\ - \frac{6}{35} \cdot 0 & - \frac{6}{35} (- \frac{35}{6}) & - \frac{6}{35} (- \frac{70}{3}) \end{array}]$

Etapa 2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{6} & \frac{19}{3} \\ 0 & 1 & 4 \end{array}]$

Etapa 2.4

Perform the row operation $R_{1} = R_{1} + \frac{1}{6} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 2.4.1

Perform the row operation $R_{1} = R_{1} + \frac{1}{6} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{1}{6} \cdot 0 & - \frac{1}{6} + \frac{1}{6} \cdot 1 & \frac{19}{3} + \frac{1}{6} \cdot 4 \\ 0 & 1 & 4 \end{array}]$

Etapa 2.4.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$m = 7$

$n = 4$

Etapa 4

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

$(7, 4)$