Matemática discreta Exemplos

$27 x \cdot 1 + 8 x \cdot 2 + 43 x \cdot 3 = 29$ , $15 x \cdot 1 - 19 x \cdot 2 + 31 x \cdot 3 = 16$ , $22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1

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

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Etapa 1.1

Simplifique cada termo.

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Etapa 1.1.1

Multiplique $27$ por $1$ .

$27 x + 8 x \cdot 2 + 43 x \cdot 3 = 29$

$15 x \cdot 1 - 19 x \cdot 2 + 31 x \cdot 3 = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.1.2

Multiplique $2$ por $8$ .

$27 x + 16 x + 43 x \cdot 3 = 29$

$15 x \cdot 1 - 19 x \cdot 2 + 31 x \cdot 3 = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.1.3

Multiplique $3$ por $43$ .

$27 x + 16 x + 129 x = 29$

$15 x \cdot 1 - 19 x \cdot 2 + 31 x \cdot 3 = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

$27 x + 16 x + 129 x = 29$

$15 x \cdot 1 - 19 x \cdot 2 + 31 x \cdot 3 = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.2

Some $27 x$ e $16 x$ .

$43 x + 129 x = 29$

$15 x \cdot 1 - 19 x \cdot 2 + 31 x \cdot 3 = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.3

Some $43 x$ e $129 x$ .

$172 x = 29$

$15 x \cdot 1 - 19 x \cdot 2 + 31 x \cdot 3 = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.4

Simplifique cada termo.

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Etapa 1.4.1

Multiplique $15$ por $1$ .

$172 x = 29$

$15 x - 19 x \cdot 2 + 31 x \cdot 3 = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.4.2

Multiplique $2$ por $- 19$ .

$172 x = 29$

$15 x - 38 x + 31 x \cdot 3 = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.4.3

Multiplique $3$ por $31$ .

$172 x = 29$

$15 x - 38 x + 93 x = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

$172 x = 29$

$15 x - 38 x + 93 x = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.5

Subtraia $38 x$ de $15 x$ .

$172 x = 29$

$- 23 x + 93 x = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.6

Some $- 23 x$ e $93 x$ .

$172 x = 29$

$70 x = 16$

$22 x \cdot 1 + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.7

Simplifique cada termo.

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Etapa 1.7.1

Multiplique $22$ por $1$ .

$172 x = 29$

$70 x = 16$

$22 x + 18 x \cdot 2 - 12 x \cdot 3 = - 17$

Etapa 1.7.2

Multiplique $2$ por $18$ .

$172 x = 29$

$70 x = 16$

$22 x + 36 x - 12 x \cdot 3 = - 17$

Etapa 1.7.3

Multiplique $3$ por $- 12$ .

$172 x = 29$

$70 x = 16$

$22 x + 36 x - 36 x = - 17$

$172 x = 29$

$70 x = 16$

$22 x + 36 x - 36 x = - 17$

Etapa 1.8

Combine os termos opostos em $22 x + 36 x - 36 x$ .

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Etapa 1.8.1

Subtraia $36 x$ de $36 x$ .

$172 x = 29$

$70 x = 16$

$22 x + 0 = - 17$

Etapa 1.8.2

Some $22 x$ e $0$ .

$172 x = 29$

$70 x = 16$

$22 x = - 17$

$172 x = 29$

$70 x = 16$

$22 x = - 17$

$172 x = 29$

$70 x = 16$

$22 x = - 17$

Etapa 2

Write the system as a matrix.

$[\begin{array}{c} 172 & 29 \\ 70 & 16 \\ 22 & - 17 \end{array}]$

Etapa 3

Encontre a forma escalonada reduzida por linhas.

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Etapa 3.1

Multiply each element of $R_{1}$ by $\frac{1}{172}$ to make the entry at $1, 1$ a $1$ .

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Etapa 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{172}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{172}{172} & \frac{29}{172} \\ 70 & 16 \\ 22 & - 17 \end{array}]$

Etapa 3.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ 70 & 16 \\ 22 & - 17 \end{array}]$

Etapa 3.2

Perform the row operation $R_{2} = R_{2} - 70 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 3.2.1

Perform the row operation $R_{2} = R_{2} - 70 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ 70 - 70 \cdot 1 & 16 - 70 (\frac{29}{172}) \\ 22 & - 17 \end{array}]$

Etapa 3.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ 0 & \frac{361}{86} \\ 22 & - 17 \end{array}]$

Etapa 3.3

Perform the row operation $R_{3} = R_{3} - 22 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 3.3.1

Perform the row operation $R_{3} = R_{3} - 22 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ 0 & \frac{361}{86} \\ 22 - 22 \cdot 1 & - 17 - 22 (\frac{29}{172}) \end{array}]$

Etapa 3.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ 0 & \frac{361}{86} \\ 0 & - \frac{1781}{86} \end{array}]$

Etapa 3.4

Multiply each element of $R_{2}$ by $\frac{86}{361}$ to make the entry at $2, 2$ a $1$ .

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Etapa 3.4.1

Multiply each element of $R_{2}$ by $\frac{86}{361}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ \frac{86}{361} \cdot 0 & \frac{86}{361} \cdot \frac{361}{86} \\ 0 & - \frac{1781}{86} \end{array}]$

Etapa 3.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ 0 & 1 \\ 0 & - \frac{1781}{86} \end{array}]$

Etapa 3.5

Perform the row operation $R_{3} = R_{3} + \frac{1781}{86} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 3.5.1

Perform the row operation $R_{3} = R_{3} + \frac{1781}{86} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ 0 & 1 \\ 0 + \frac{1781}{86} \cdot 0 & - \frac{1781}{86} + \frac{1781}{86} \cdot 1 \end{array}]$

Etapa 3.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{29}{172} \\ 0 & 1 \\ 0 & 0 \end{array}]$

Etapa 3.6

Perform the row operation $R_{1} = R_{1} - \frac{29}{172} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 3.6.1

Perform the row operation $R_{1} = R_{1} - \frac{29}{172} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{29}{172} \cdot 0 & \frac{29}{172} - \frac{29}{172} \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

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.

$(error = error)$