Matemática discreta Ejemplos

$3 x - 6 y + 6 z = - 3$ , $- 6 + 12 y - 12 z = 6$ , $6 x - 12 y + 12 z = - 6$

Paso 1

Mueve todos los términos que no contengan una variable al lado derecho de la ecuación.

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Paso 1.1

Suma $6$ a ambos lados de la ecuación.

$3 x - 6 y + 6 z = - 3$

$12 y - 12 z = 6 + 6$

$6 x - 12 y + 12 z = - 6$

Paso 1.2

Suma $6$ y $6$ .

$3 x - 6 y + 6 z = - 3$

$12 y - 12 z = 12$

$6 x - 12 y + 12 z = - 6$

$3 x - 6 y + 6 z = - 3$

$12 y - 12 z = 12$

$6 x - 12 y + 12 z = - 6$

Paso 2

Write the system as a matrix.

$[\begin{array}{c} 3 & - 6 & 6 & - 3 \\ 0 & 12 & - 12 & 12 \\ 6 & - 12 & 12 & - 6 \end{array}]$

Paso 3

Obtén la forma escalonada reducida por filas.

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Paso 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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Paso 3.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 6}{3} & \frac{6}{3} & \frac{- 3}{3} \\ 0 & 12 & - 12 & 12 \\ 6 & - 12 & 12 & - 6 \end{array}]$

Paso 3.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & 2 & - 1 \\ 0 & 12 & - 12 & 12 \\ 6 & - 12 & 12 & - 6 \end{array}]$

Paso 3.2

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 3.2.1

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 2 & - 1 \\ 0 & 12 & - 12 & 12 \\ 6 - 6 \cdot 1 & - 12 - 6 \cdot - 2 & 12 - 6 \cdot 2 & - 6 - 6 \cdot - 1 \end{array}]$

Paso 3.2.2

Simplifica $R_{3}$ .

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

Paso 3.3

Multiply each element of $R_{2}$ by $\frac{1}{12}$ to make the entry at $2, 2$ a $1$ .

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Paso 3.3.1

Multiply each element of $R_{2}$ by $\frac{1}{12}$ to make the entry at $2, 2$ a $1$ .

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

Paso 3.3.2

Simplifica $R_{2}$ .

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

Paso 3.4

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

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Paso 3.4.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 2 + 2 \cdot - 1 & - 1 + 2 \cdot 1 \\ 0 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}]$

Paso 3.4.2

Simplifica $R_{1}$ .

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

Paso 4

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

$x = 1$

$y - z = 1$

$0 = 0$

Paso 5

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

$(1, 1 + z, z)$