Matemática discreta Ejemplos

$x + 4 y = 0$ , $x + 5 y + z = 1$ , $- 4 x - y - z = - 17$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 4 & 0 & 0 \\ 1 & 5 & 1 & 1 \\ - 4 & - 1 & - 1 & - 17 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

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

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Paso 2.1.1

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

$[\begin{array}{c} 1 & 4 & 0 & 0 \\ 1 - 1 & 5 - 4 & 1 - 0 & 1 - 0 \\ - 4 & - 1 & - 1 & - 17 \end{array}]$

Paso 2.1.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 4 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ - 4 & - 1 & - 1 & - 17 \end{array}]$

Paso 2.2

Perform the row operation $R_{3} = R_{3} + 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 2.2.1

Perform the row operation $R_{3} = R_{3} + 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 4 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ - 4 + 4 \cdot 1 & - 1 + 4 \cdot 4 & - 1 + 4 \cdot 0 & - 17 + 4 \cdot 0 \end{array}]$

Paso 2.2.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 4 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 15 & - 1 & - 17 \end{array}]$

Paso 2.3

Perform the row operation $R_{3} = R_{3} - 15 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 2.3.1

Perform the row operation $R_{3} = R_{3} - 15 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 4 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 - 15 \cdot 0 & 15 - 15 \cdot 1 & - 1 - 15 \cdot 1 & - 17 - 15 \cdot 1 \end{array}]$

Paso 2.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 4 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & - 16 & - 32 \end{array}]$

Paso 2.4

Multiply each element of $R_{3}$ by $- \frac{1}{16}$ to make the entry at $3, 3$ a $1$ .

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Paso 2.4.1

Multiply each element of $R_{3}$ by $- \frac{1}{16}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 4 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ - \frac{1}{16} \cdot 0 & - \frac{1}{16} \cdot 0 & - \frac{1}{16} \cdot - 16 & - \frac{1}{16} \cdot - 32 \end{array}]$

Paso 2.4.2

Simplifica $R_{3}$ .

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

Paso 2.5

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

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Paso 2.5.1

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

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

Paso 2.5.2

Simplifica $R_{2}$ .

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

Paso 2.6

Perform the row operation $R_{1} = R_{1} - 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Paso 2.6.1

Perform the row operation $R_{1} = R_{1} - 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Paso 2.6.2

Simplifica $R_{1}$ .

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

Paso 3

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

$x = 4$

$y = - 1$

$z = 2$

Paso 4

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

$(4, - 1, 2)$