Álgebra lineal Ejemplos

$5 x - 6 y = 0$ , $2 x - y = 17$ , $3 x - 7 y = 0$

Paso 1

Escribe el sistema de ecuaciones en forma de matriz.

$[\begin{array}{c} 5 & - 6 & 0 \\ 2 & - 1 & 17 \\ 3 & - 7 & 0 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

Multiply each element of $R_{1}$ by $\frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

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Paso 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{5}{5} & \frac{- 6}{5} & \frac{0}{5} \\ 2 & - 1 & 17 \\ 3 & - 7 & 0 \end{array}]$

Paso 2.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 2 & - 1 & 17 \\ 3 & - 7 & 0 \end{array}]$

Paso 2.2

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 2.2.1

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

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 2 - 2 \cdot 1 & - 1 - 2 (- \frac{6}{5}) & 17 - 2 \cdot 0 \\ 3 & - 7 & 0 \end{array}]$

Paso 2.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & \frac{7}{5} & 17 \\ 3 & - 7 & 0 \end{array}]$

Paso 2.3

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

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Paso 2.3.1

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

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & \frac{7}{5} & 17 \\ 3 - 3 \cdot 1 & - 7 - 3 (- \frac{6}{5}) & 0 - 3 \cdot 0 \end{array}]$

Paso 2.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & \frac{7}{5} & 17 \\ 0 & - \frac{17}{5} & 0 \end{array}]$

Paso 2.4

Multiply each element of $R_{2}$ by $\frac{5}{7}$ to make the entry at $2, 2$ a $1$ .

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Paso 2.4.1

Multiply each element of $R_{2}$ by $\frac{5}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ \frac{5}{7} \cdot 0 & \frac{5}{7} \cdot \frac{7}{5} & \frac{5}{7} \cdot 17 \\ 0 & - \frac{17}{5} & 0 \end{array}]$

Paso 2.4.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & 1 & \frac{85}{7} \\ 0 & - \frac{17}{5} & 0 \end{array}]$

Paso 2.5

Perform the row operation $R_{3} = R_{3} + \frac{17}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 2.5.1

Perform the row operation $R_{3} = R_{3} + \frac{17}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & 1 & \frac{85}{7} \\ 0 + \frac{17}{5} \cdot 0 & - \frac{17}{5} + \frac{17}{5} \cdot 1 & 0 + \frac{17}{5} \cdot \frac{85}{7} \end{array}]$

Paso 2.5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & 1 & \frac{85}{7} \\ 0 & 0 & \frac{289}{7} \end{array}]$

Paso 2.6

Multiply each element of $R_{3}$ by $\frac{7}{289}$ to make the entry at $3, 3$ a $1$ .

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Paso 2.6.1

Multiply each element of $R_{3}$ by $\frac{7}{289}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & 1 & \frac{85}{7} \\ \frac{7}{289} \cdot 0 & \frac{7}{289} \cdot 0 & \frac{7}{289} \cdot \frac{289}{7} \end{array}]$

Paso 2.6.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & 1 & \frac{85}{7} \\ 0 & 0 & 1 \end{array}]$

Paso 2.7

Perform the row operation $R_{2} = R_{2} - \frac{85}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Paso 2.7.1

Perform the row operation $R_{2} = R_{2} - \frac{85}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 - \frac{85}{7} \cdot 0 & 1 - \frac{85}{7} \cdot 0 & \frac{85}{7} - \frac{85}{7} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Paso 2.7.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Paso 2.8

Perform the row operation $R_{1} = R_{1} + \frac{6}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Paso 2.8.1

Perform the row operation $R_{1} = R_{1} + \frac{6}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{6}{5} \cdot 0 & - \frac{6}{5} + \frac{6}{5} \cdot 1 & 0 + \frac{6}{5} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Paso 2.8.2

Simplifica $R_{1}$ .

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

Paso 3

Usa la matriz de resultados para declarar las soluciones finales en el sistema de ecuaciones.

$x = 0$

$y = 0$

$0 = 1$

Paso 4

Como $0 \neq 1$ , no hay soluciones.

No hay solución