Matemática discreta Ejemplos

$x + y + 5 z = 2$ , $5 x + 5 y + 25 z = 10$ , $8 x + 8 y + 40 z = 16$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 5 & 2 \\ 5 & 5 & 25 & 10 \\ 8 & 8 & 40 & 16 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 2.1.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 5 & 2 \\ 5 - 5 \cdot 1 & 5 - 5 \cdot 1 & 25 - 5 \cdot 5 & 10 - 5 \cdot 2 \\ 8 & 8 & 40 & 16 \end{array}]$

Paso 2.1.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 5 & 2 \\ 0 & 0 & 0 & 0 \\ 8 & 8 & 40 & 16 \end{array}]$

Paso 2.2

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 2.2.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 5 & 2 \\ 0 & 0 & 0 & 0 \\ 8 - 8 \cdot 1 & 8 - 8 \cdot 1 & 40 - 8 \cdot 5 & 16 - 8 \cdot 2 \end{array}]$

Paso 2.2.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 5 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Paso 3

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

$x + y + 5 z = 2$

$0 = 0$

Paso 4

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

$(2 - y - 5 z, y, z)$