Matemática discreta Exemplos

$2 x + 3 y + u = 12$ , $2 x + y + v = 8$ , $- 3 x - 2 y + p = 0$

Etapa 1

Move variables to the left and constant terms to the right.

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Etapa 1.1

Mova $3 y$ .

$2 x + u + 3 y = 12$

$2 x + y + v = 8$

$- 3 x - 2 y + p = 0$

Etapa 1.2

Reordene $2 x$ e $u$ .

$u + 2 x + 3 y = 12$

$2 x + y + v = 8$

$- 3 x - 2 y + p = 0$

Etapa 1.3

Mova $y$ .

$u + 2 x + 3 y = 12$

$2 x + v + y = 8$

$- 3 x - 2 y + p = 0$

Etapa 1.4

Reordene $2 x$ e $v$ .

$u + 2 x + 3 y = 12$

$v + 2 x + y = 8$

$- 3 x - 2 y + p = 0$

Etapa 1.5

Mova $- 2 y$ .

$u + 2 x + 3 y = 12$

$v + 2 x + y = 8$

$- 3 x + p - 2 y = 0$

Etapa 1.6

Reordene $- 3 x$ e $p$ .

$u + 2 x + 3 y = 12$

$v + 2 x + y = 8$

$p - 3 x - 2 y = 0$

$u + 2 x + 3 y = 12$

$v + 2 x + y = 8$

$p - 3 x - 2 y = 0$

Etapa 2

Write the system as a matrix.

$[\begin{array}{c} 0 & 1 & 0 & 2 & 3 & 12 \\ 0 & 0 & 1 & 2 & 1 & 8 \\ 1 & 0 & 0 & - 3 & - 2 & 0 \end{array}]$

Etapa 3

Encontre a forma escalonada reduzida por linhas.

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Etapa 3.1

Swap $R_{3}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 1 & 0 & 0 & - 3 & - 2 & 0 \\ 0 & 0 & 1 & 2 & 1 & 8 \\ 0 & 1 & 0 & 2 & 3 & 12 \end{array}]$

Etapa 3.2

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 0 & 0 & - 3 & - 2 & 0 \\ 0 & 1 & 0 & 2 & 3 & 12 \\ 0 & 0 & 1 & 2 & 1 & 8 \end{array}]$

Etapa 4

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

$p - 3 x - 2 y = 0$

$u + 2 x + 3 y = 12$

$v + 2 x + y = 8$

Etapa 5

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

$(3 x + 2 y, 12 - 2 x - 3 y, 8 - 2 x - y, x, y)$