Matemática discreta Exemplos

$x + 3 y = 9$ , $3 x - y = 7$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 3 & 9 \\ 3 & - 1 & 7 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

$[\begin{array}{c} 1 & 3 & 9 \\ 3 - 3 \cdot 1 & - 1 - 3 \cdot 3 & 7 - 3 \cdot 9 \end{array}]$

Etapa 2.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 3 & 9 \\ 0 & - 10 & - 20 \end{array}]$

Etapa 2.2

Multiply each element of $R_{2}$ by $- \frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.2.1

Multiply each element of $R_{2}$ by $- \frac{1}{10}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 3 & 9 \\ - \frac{1}{10} \cdot 0 & - \frac{1}{10} \cdot - 10 & - \frac{1}{10} \cdot - 20 \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 3 & 9 \\ 0 & 1 & 2 \end{array}]$

Etapa 2.3

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

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Etapa 2.3.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & 9 - 3 \cdot 2 \\ 0 & 1 & 2 \end{array}]$

Etapa 2.3.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$x = 3$

$y = 2$

Etapa 4

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

$(3, 2)$