Matemática discreta Ejemplos

$x + 2 y - 3 z = 4$ , $2 x + 5 y + 2 z = 18$ , $7 x + 16 y - 5 z = 48$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 2 & - 3 & 4 \\ 2 & 5 & 2 & 18 \\ 7 & 16 & - 5 & 48 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

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

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Paso 2.1.1

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

$[\begin{array}{c} 1 & 2 & - 3 & 4 \\ 2 - 2 \cdot 1 & 5 - 2 \cdot 2 & 2 - 2 \cdot - 3 & 18 - 2 \cdot 4 \\ 7 & 16 & - 5 & 48 \end{array}]$

Paso 2.1.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - 3 & 4 \\ 0 & 1 & 8 & 10 \\ 7 & 16 & - 5 & 48 \end{array}]$

Paso 2.2

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 2.2.1

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & - 3 & 4 \\ 0 & 1 & 8 & 10 \\ 7 - 7 \cdot 1 & 16 - 7 \cdot 2 & - 5 - 7 \cdot - 3 & 48 - 7 \cdot 4 \end{array}]$

Paso 2.2.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 2 & - 3 & 4 \\ 0 & 1 & 8 & 10 \\ 0 & 2 & 16 & 20 \end{array}]$

Paso 2.3

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

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Paso 2.3.1

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

$[\begin{array}{c} 1 & 2 & - 3 & 4 \\ 0 & 1 & 8 & 10 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & 16 - 2 \cdot 8 & 20 - 2 \cdot 10 \end{array}]$

Paso 2.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 2 & - 3 & 4 \\ 0 & 1 & 8 & 10 \\ 0 & 0 & 0 & 0 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 3 - 2 \cdot 8 & 4 - 2 \cdot 10 \\ 0 & 1 & 8 & 10 \\ 0 & 0 & 0 & 0 \end{array}]$

Paso 2.4.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 19 & - 16 \\ 0 & 1 & 8 & 10 \\ 0 & 0 & 0 & 0 \end{array}]$

Paso 3

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

$x - 19 z = - 16$

$y + 8 z = 10$

$0 = 0$

Paso 4

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

$(- 16 + 19 z, 10 - 8 z, z)$