Matemática discreta Exemplos

$x + y = 11$ , $5 x + 4 y = 49$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 11 \\ 5 & 4 & 49 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

$[\begin{array}{c} 1 & 1 & 11 \\ 5 - 5 \cdot 1 & 4 - 5 \cdot 1 & 49 - 5 \cdot 11 \end{array}]$

Etapa 2.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 11 \\ 0 & - 1 & - 6 \end{array}]$

Etapa 2.2

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

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Etapa 2.2.1

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

$[\begin{array}{c} 1 & 1 & 11 \\ - 0 & - - 1 & - - 6 \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 11 \\ 0 & 1 & 6 \end{array}]$

Etapa 2.3

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

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Etapa 2.3.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 11 - 6 \\ 0 & 1 & 6 \end{array}]$

Etapa 2.3.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$x = 5$

$y = 6$

Etapa 4

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

$(5, 6)$