Matemática discreta Exemplos

$z = 3 x + 9 y$ , $x - y = - 10$ , $3 x + 2 y = 30$

Etapa 1

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

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Etapa 1.1

Mova todos os termos que contêm variáveis para o lado esquerdo da equação.

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Etapa 1.1.1

Subtraia $3 x$ dos dois lados da equação.

$z - 3 x = 9 y$

$x - y = - 10$

$3 x + 2 y = 30$

Etapa 1.1.2

Subtraia $9 y$ dos dois lados da equação.

$z - 3 x - 9 y = 0$

$x - y = - 10$

$3 x + 2 y = 30$

$z - 3 x - 9 y = 0$

$x - y = - 10$

$3 x + 2 y = 30$

Etapa 1.2

Mova $z$ .

$- 3 x - 9 y + z = 0$

$x - y = - 10$

$3 x + 2 y = 30$

$- 3 x - 9 y + z = 0$

$x - y = - 10$

$3 x + 2 y = 30$

Etapa 2

Write the system as a matrix.

$[\begin{array}{c} - 3 & - 9 & 1 & 0 \\ 1 & - 1 & 0 & - 10 \\ 3 & 2 & 0 & 30 \end{array}]$

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 $1$ .

$[\begin{array}{c} - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 9 & - \frac{1}{3} \cdot 1 & - \frac{1}{3} \cdot 0 \\ 1 & - 1 & 0 & - 10 \\ 3 & 2 & 0 & 30 \end{array}]$

Etapa 3.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 1 & - 1 & 0 & - 10 \\ 3 & 2 & 0 & 30 \end{array}]$

Etapa 3.2

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

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Etapa 3.2.1

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

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 1 - 1 & - 1 - 3 & 0 + \frac{1}{3} & - 10 - 0 \\ 3 & 2 & 0 & 30 \end{array}]$

Etapa 3.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & - 4 & \frac{1}{3} & - 10 \\ 3 & 2 & 0 & 30 \end{array}]$

Etapa 3.3

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

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Etapa 3.3.1

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

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & - 4 & \frac{1}{3} & - 10 \\ 3 - 3 \cdot 1 & 2 - 3 \cdot 3 & 0 - 3 (- \frac{1}{3}) & 30 - 3 \cdot 0 \end{array}]$

Etapa 3.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & - 4 & \frac{1}{3} & - 10 \\ 0 & - 7 & 1 & 30 \end{array}]$

Etapa 3.4

Multiply each element of $R_{2}$ by $- \frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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Etapa 3.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot - 4 & - \frac{1}{4} \cdot \frac{1}{3} & - \frac{1}{4} \cdot - 10 \\ 0 & - 7 & 1 & 30 \end{array}]$

Etapa 3.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & 1 & - \frac{1}{12} & \frac{5}{2} \\ 0 & - 7 & 1 & 30 \end{array}]$

Etapa 3.5

Perform the row operation $R_{3} = R_{3} + 7 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 3.5.1

Perform the row operation $R_{3} = R_{3} + 7 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & 1 & - \frac{1}{12} & \frac{5}{2} \\ 0 + 7 \cdot 0 & - 7 + 7 \cdot 1 & 1 + 7 (- \frac{1}{12}) & 30 + 7 (\frac{5}{2}) \end{array}]$

Etapa 3.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & 1 & - \frac{1}{12} & \frac{5}{2} \\ 0 & 0 & \frac{5}{12} & \frac{95}{2} \end{array}]$

Etapa 3.6

Multiply each element of $R_{3}$ by $\frac{12}{5}$ to make the entry at $3, 3$ a $1$ .

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Etapa 3.6.1

Multiply each element of $R_{3}$ by $\frac{12}{5}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & 1 & - \frac{1}{12} & \frac{5}{2} \\ \frac{12}{5} \cdot 0 & \frac{12}{5} \cdot 0 & \frac{12}{5} \cdot \frac{5}{12} & \frac{12}{5} \cdot \frac{95}{2} \end{array}]$

Etapa 3.6.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & 1 & - \frac{1}{12} & \frac{5}{2} \\ 0 & 0 & 1 & 114 \end{array}]$

Etapa 3.7

Perform the row operation $R_{2} = R_{2} + \frac{1}{12} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 3.7.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{12} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 + \frac{1}{12} \cdot 0 & 1 + \frac{1}{12} \cdot 0 & - \frac{1}{12} + \frac{1}{12} \cdot 1 & \frac{5}{2} + \frac{1}{12} \cdot 114 \\ 0 & 0 & 1 & 114 \end{array}]$

Etapa 3.7.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - \frac{1}{3} & 0 \\ 0 & 1 & 0 & 12 \\ 0 & 0 & 1 & 114 \end{array}]$

Etapa 3.8

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

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Etapa 3.8.1

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

$[\begin{array}{c} 1 + \frac{1}{3} \cdot 0 & 3 + \frac{1}{3} \cdot 0 & - \frac{1}{3} + \frac{1}{3} \cdot 1 & 0 + \frac{1}{3} \cdot 114 \\ 0 & 1 & 0 & 12 \\ 0 & 0 & 1 & 114 \end{array}]$

Etapa 3.8.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 3 & 0 & 38 \\ 0 & 1 & 0 & 12 \\ 0 & 0 & 1 & 114 \end{array}]$

Etapa 3.9

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

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Etapa 3.9.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & 0 - 3 \cdot 0 & 38 - 3 \cdot 12 \\ 0 & 1 & 0 & 12 \\ 0 & 0 & 1 & 114 \end{array}]$

Etapa 3.9.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 12 \\ 0 & 0 & 1 & 114 \end{array}]$

Etapa 4

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

$x = 2$

$y = 12$

$z = 114$

Etapa 5

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

$(2, 12, 114)$