Matemática discreta Exemplos

$3 x - 4 y = 9$ , $- 2 x + 3 y = - 6$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 3 & - 4 & 9 \\ - 2 & 3 & - 6 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

Multiply each element of $R_{1}$ by $\frac{1}{3}$ 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 $1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{- 4}{3} & \frac{9}{3} \\ - 2 & 3 & - 6 \end{array}]$

Etapa 2.1.2

Simplifique $R_{1}$ .

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

Etapa 2.2

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 2.2.1

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

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

Etapa 2.2.2

Simplifique $R_{2}$ .

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

Etapa 2.3

Multiply each element of $R_{2}$ by $3$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.3.1

Multiply each element of $R_{2}$ by $3$ to make the entry at $2, 2$ a $1$ .

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

Etapa 2.3.2

Simplifique $R_{2}$ .

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

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 $0$ .

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

Etapa 2.4.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$x = 3$

$y = 0$

Etapa 4

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

$(3, 0)$