Matemática discreta Exemplos

$3 x - 4 y = 16$ , $6 x + 8 y = - 2$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 3 & - 4 & 16 \\ 6 & 8 & - 2 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 4}{3} & \frac{16}{3} \\ 6 & 8 & - 2 \end{array}]$

Etapa 2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{4}{3} & \frac{16}{3} \\ 6 & 8 & - 2 \end{array}]$

Etapa 2.2

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

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Etapa 2.2.1

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

$[\begin{array}{c} 1 & - \frac{4}{3} & \frac{16}{3} \\ 6 - 6 \cdot 1 & 8 - 6 (- \frac{4}{3}) & - 2 - 6 (\frac{16}{3}) \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{4}{3} & \frac{16}{3} \\ 0 & 16 & - 34 \end{array}]$

Etapa 2.3

Multiply each element of $R_{2}$ by $\frac{1}{16}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{16}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{4}{3} & \frac{16}{3} \\ \frac{0}{16} & \frac{16}{16} & \frac{- 34}{16} \end{array}]$

Etapa 2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{4}{3} & \frac{16}{3} \\ 0 & 1 & - \frac{17}{8} \end{array}]$

Etapa 2.4

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

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Etapa 2.4.1

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

$[\begin{array}{c} 1 + \frac{4}{3} \cdot 0 & - \frac{4}{3} + \frac{4}{3} \cdot 1 & \frac{16}{3} + \frac{4}{3} (- \frac{17}{8}) \\ 0 & 1 & - \frac{17}{8} \end{array}]$

Etapa 2.4.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{2} \\ 0 & 1 & - \frac{17}{8} \end{array}]$

Etapa 3

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

$x = \frac{5}{2}$

$y = - \frac{17}{8}$

Etapa 4

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

$(\frac{5}{2}, - \frac{17}{8})$