Matemática discreta Ejemplos

$x - 2 y + z = 0$ , $y - 4 z = 4$ , $- 3 y + 13 z = - 9$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 2 & 1 & 0 \\ 0 & 1 & - 4 & 4 \\ 0 & - 3 & 13 & - 9 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

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

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Paso 2.1.1

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

$[\begin{array}{c} 1 & - 2 & 1 & 0 \\ 0 & 1 & - 4 & 4 \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 13 + 3 \cdot - 4 & - 9 + 3 \cdot 4 \end{array}]$

Paso 2.1.2

Simplifica $R_{3}$ .

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

Paso 2.2

Perform the row operation $R_{2} = R_{2} + 4 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Paso 2.2.1

Perform the row operation $R_{2} = R_{2} + 4 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 1 & 0 \\ 0 + 4 \cdot 0 & 1 + 4 \cdot 0 & - 4 + 4 \cdot 1 & 4 + 4 \cdot 3 \\ 0 & 0 & 1 & 3 \end{array}]$

Paso 2.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 1 & 0 \\ 0 & 1 & 0 & 16 \\ 0 & 0 & 1 & 3 \end{array}]$

Paso 2.3

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

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Paso 2.3.1

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

$[\begin{array}{c} 1 - 0 & - 2 - 0 & 1 - 1 & 0 - 3 \\ 0 & 1 & 0 & 16 \\ 0 & 0 & 1 & 3 \end{array}]$

Paso 2.3.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & 0 & - 3 \\ 0 & 1 & 0 & 16 \\ 0 & 0 & 1 & 3 \end{array}]$

Paso 2.4

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 2.4.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & - 3 + 2 \cdot 16 \\ 0 & 1 & 0 & 16 \\ 0 & 0 & 1 & 3 \end{array}]$

Paso 2.4.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 29 \\ 0 & 1 & 0 & 16 \\ 0 & 0 & 1 & 3 \end{array}]$

Paso 3

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

$x = 29$

$y = 16$

$z = 3$

Paso 4

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

$(29, 16, 3)$