Matemática discreta Exemplos

$2 x \cdot 1 - x \cdot 2 - x \cdot 3 = 1$ , $3 x \cdot 1 + 2 x \cdot 2 + x \cdot 3 = 12$ , $x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1

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

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Etapa 1.1

Combine os termos opostos em $2 x \cdot 1 - x \cdot 2 - x \cdot 3$ .

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Etapa 1.1.1

Reorganize os fatores nos termos $2 x \cdot 1$ e $- x \cdot 2$ .

$1 \cdot 2 x - 1 \cdot 2 x - x \cdot 3 = 1$

$3 x \cdot 1 + 2 x \cdot 2 + x \cdot 3 = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.1.2

Subtraia $2 x$ de $1 \cdot 2 x$ .

$0 - x \cdot 3 = 1$

$3 x \cdot 1 + 2 x \cdot 2 + x \cdot 3 = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

$0 - x \cdot 3 = 1$

$3 x \cdot 1 + 2 x \cdot 2 + x \cdot 3 = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.2

Subtraia $x \cdot 3$ de $0$ .

$- x \cdot 3 = 1$

$3 x \cdot 1 + 2 x \cdot 2 + x \cdot 3 = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.3

Multiplique $3$ por $- 1$ .

$- 3 x = 1$

$3 x \cdot 1 + 2 x \cdot 2 + x \cdot 3 = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.4

Simplifique cada termo.

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Etapa 1.4.1

Multiplique $3$ por $1$ .

$- 3 x = 1$

$3 x + 2 x \cdot 2 + x \cdot 3 = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.4.2

Multiplique $2$ por $2$ .

$- 3 x = 1$

$3 x + 4 x + x \cdot 3 = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.4.3

Mova $3$ para a esquerda de $x$ .

$- 3 x = 1$

$3 x + 4 x + 3 x = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

$- 3 x = 1$

$3 x + 4 x + 3 x = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.5

Some $3 x$ e $4 x$ .

$- 3 x = 1$

$7 x + 3 x = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.6

Some $7 x$ e $3 x$ .

$- 3 x = 1$

$10 x = 12$

$x \cdot 1 + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.7

Simplifique cada termo.

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Etapa 1.7.1

Multiplique $x$ por $1$ .

$- 3 x = 1$

$10 x = 12$

$x + 2 x \cdot 2 + 2 x \cdot 3 = 8$

Etapa 1.7.2

Multiplique $2$ por $2$ .

$- 3 x = 1$

$10 x = 12$

$x + 4 x + 2 x \cdot 3 = 8$

Etapa 1.7.3

Multiplique $3$ por $2$ .

$- 3 x = 1$

$10 x = 12$

$x + 4 x + 6 x = 8$

$- 3 x = 1$

$10 x = 12$

$x + 4 x + 6 x = 8$

Etapa 1.8

Some $x$ e $4 x$ .

$- 3 x = 1$

$10 x = 12$

$5 x + 6 x = 8$

Etapa 1.9

Some $5 x$ e $6 x$ .

$- 3 x = 1$

$10 x = 12$

$11 x = 8$

$- 3 x = 1$

$10 x = 12$

$11 x = 8$

Etapa 2

Write the system as a matrix.

$[\begin{array}{c} - 3 & 1 \\ 10 & 12 \\ 11 & 8 \end{array}]$

Etapa 3

Encontre a forma escalonada reduzida por linhas.

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Etapa 3.1

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

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Etapa 3.1.1

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

$[\begin{array}{c} - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 1 \\ 10 & 12 \\ 11 & 8 \end{array}]$

Etapa 3.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} \\ 10 & 12 \\ 11 & 8 \end{array}]$

Etapa 3.2

Perform the row operation $R_{2} = R_{2} - 10 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 3.2.1

Perform the row operation $R_{2} = R_{2} - 10 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{3} \\ 10 - 10 \cdot 1 & 12 - 10 (- \frac{1}{3}) \\ 11 & 8 \end{array}]$

Etapa 3.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} \\ 0 & \frac{46}{3} \\ 11 & 8 \end{array}]$

Etapa 3.3

Perform the row operation $R_{3} = R_{3} - 11 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 3.3.1

Perform the row operation $R_{3} = R_{3} - 11 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{3} \\ 0 & \frac{46}{3} \\ 11 - 11 \cdot 1 & 8 - 11 (- \frac{1}{3}) \end{array}]$

Etapa 3.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} \\ 0 & \frac{46}{3} \\ 0 & \frac{35}{3} \end{array}]$

Etapa 3.4

Multiply each element of $R_{2}$ by $\frac{3}{46}$ to make the entry at $2, 2$ a $1$ .

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Etapa 3.4.1

Multiply each element of $R_{2}$ by $\frac{3}{46}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{3} \\ \frac{3}{46} \cdot 0 & \frac{3}{46} \cdot \frac{46}{3} \\ 0 & \frac{35}{3} \end{array}]$

Etapa 3.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} \\ 0 & 1 \\ 0 & \frac{35}{3} \end{array}]$

Etapa 3.5

Perform the row operation $R_{3} = R_{3} - \frac{35}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 3.5.1

Perform the row operation $R_{3} = R_{3} - \frac{35}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{3} \\ 0 & 1 \\ 0 - \frac{35}{3} \cdot 0 & \frac{35}{3} - \frac{35}{3} \cdot 1 \end{array}]$

Etapa 3.5.2

Simplifique $R_{3}$ .

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

Etapa 3.6

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

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Etapa 3.6.1

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

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

Etapa 3.6.2

Simplifique $R_{1}$ .

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

Etapa 4

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

$x = 0$

$0 = 1$

$0 = 0$

Etapa 5

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

$(error = error)$