Matemática discreta Exemplos

$9 x \cdot 1 + 29 x \cdot 2 + 3 x \cdot 3 = 12$ , $4 x \cdot 1 + 13 x \cdot 2 + 4 x \cdot 3 = - 7$ , $x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

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 $9$ por $1$ .

$9 x + 29 x \cdot 2 + 3 x \cdot 3 = 12$

$4 x \cdot 1 + 13 x \cdot 2 + 4 x \cdot 3 = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.1.2

Multiplique $2$ por $29$ .

$9 x + 58 x + 3 x \cdot 3 = 12$

$4 x \cdot 1 + 13 x \cdot 2 + 4 x \cdot 3 = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.1.3

Multiplique $3$ por $3$ .

$9 x + 58 x + 9 x = 12$

$4 x \cdot 1 + 13 x \cdot 2 + 4 x \cdot 3 = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

$9 x + 58 x + 9 x = 12$

$4 x \cdot 1 + 13 x \cdot 2 + 4 x \cdot 3 = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.2

Some $9 x$ e $58 x$ .

$67 x + 9 x = 12$

$4 x \cdot 1 + 13 x \cdot 2 + 4 x \cdot 3 = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.3

Some $67 x$ e $9 x$ .

$76 x = 12$

$4 x \cdot 1 + 13 x \cdot 2 + 4 x \cdot 3 = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.4

Simplifique cada termo.

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Etapa 1.4.1

Multiplique $4$ por $1$ .

$76 x = 12$

$4 x + 13 x \cdot 2 + 4 x \cdot 3 = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.4.2

Multiplique $2$ por $13$ .

$76 x = 12$

$4 x + 26 x + 4 x \cdot 3 = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.4.3

Multiplique $3$ por $4$ .

$76 x = 12$

$4 x + 26 x + 12 x = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

$76 x = 12$

$4 x + 26 x + 12 x = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.5

Some $4 x$ e $26 x$ .

$76 x = 12$

$30 x + 12 x = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.6

Some $30 x$ e $12 x$ .

$76 x = 12$

$42 x = - 7$

$x \cdot 1 + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.7

Simplifique cada termo.

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Etapa 1.7.1

Multiplique $x$ por $1$ .

$76 x = 12$

$42 x = - 7$

$x + 3 x \cdot 2 - 5 x \cdot 3 = 1$

Etapa 1.7.2

Multiplique $2$ por $3$ .

$76 x = 12$

$42 x = - 7$

$x + 6 x - 5 x \cdot 3 = 1$

Etapa 1.7.3

Multiplique $3$ por $- 5$ .

$76 x = 12$

$42 x = - 7$

$x + 6 x - 15 x = 1$

$76 x = 12$

$42 x = - 7$

$x + 6 x - 15 x = 1$

Etapa 1.8

Some $x$ e $6 x$ .

$76 x = 12$

$42 x = - 7$

$7 x - 15 x = 1$

Etapa 1.9

Subtraia $15 x$ de $7 x$ .

$76 x = 12$

$42 x = - 7$

$- 8 x = 1$

$76 x = 12$

$42 x = - 7$

$- 8 x = 1$

Etapa 2

Write the system as a matrix.

$[\begin{array}{c} 76 & 12 \\ 42 & - 7 \\ - 8 & 1 \end{array}]$

Etapa 3

Encontre a forma escalonada reduzida por linhas.

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Etapa 3.1

Multiply each element of $R_{1}$ by $\frac{1}{76}$ to make the entry at $1, 1$ a $1$ .

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Etapa 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{76}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{76}{76} & \frac{12}{76} \\ 42 & - 7 \\ - 8 & 1 \end{array}]$

Etapa 3.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ 42 & - 7 \\ - 8 & 1 \end{array}]$

Etapa 3.2

Perform the row operation $R_{2} = R_{2} - 42 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 3.2.1

Perform the row operation $R_{2} = R_{2} - 42 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ 42 - 42 \cdot 1 & - 7 - 42 (\frac{3}{19}) \\ - 8 & 1 \end{array}]$

Etapa 3.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ 0 & - \frac{259}{19} \\ - 8 & 1 \end{array}]$

Etapa 3.3

Perform the row operation $R_{3} = R_{3} + 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 3.3.1

Perform the row operation $R_{3} = R_{3} + 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ 0 & - \frac{259}{19} \\ - 8 + 8 \cdot 1 & 1 + 8 (\frac{3}{19}) \end{array}]$

Etapa 3.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ 0 & - \frac{259}{19} \\ 0 & \frac{43}{19} \end{array}]$

Etapa 3.4

Multiply each element of $R_{2}$ by $- \frac{19}{259}$ to make the entry at $2, 2$ a $1$ .

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Etapa 3.4.1

Multiply each element of $R_{2}$ by $- \frac{19}{259}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ - \frac{19}{259} \cdot 0 & - \frac{19}{259} (- \frac{259}{19}) \\ 0 & \frac{43}{19} \end{array}]$

Etapa 3.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ 0 & 1 \\ 0 & \frac{43}{19} \end{array}]$

Etapa 3.5

Perform the row operation $R_{3} = R_{3} - \frac{43}{19} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 3.5.1

Perform the row operation $R_{3} = R_{3} - \frac{43}{19} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ 0 & 1 \\ 0 - \frac{43}{19} \cdot 0 & \frac{43}{19} - \frac{43}{19} \cdot 1 \end{array}]$

Etapa 3.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{3}{19} \\ 0 & 1 \\ 0 & 0 \end{array}]$

Etapa 3.6

Perform the row operation $R_{1} = R_{1} - \frac{3}{19} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 3.6.1

Perform the row operation $R_{1} = R_{1} - \frac{3}{19} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{3}{19} \cdot 0 & \frac{3}{19} - \frac{3}{19} \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

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.

$(error = error)$