Álgebra Ejemplos

3 x + 1 = 0 y

$3 x + 1 = 0 y$ ,

y + 2 x > - 9

$y + 2 x > - 9$

Paso 1

Introduce las variables de holgura

u

$u$ y

v

$v$ para reemplazar las desigualdades con ecuaciones.

y + 2 x - Z = - 9

$y + 2 x - Z = - 9$

3 x + 1 = 0

$3 x + 1 = 0$

Paso 2

Resta

1

$1$ de ambos lados de la ecuación.

y + 2 x - Z = - 9, 3 x = - 1

$y + 2 x - Z = - 9, 3 x = - 1$

Paso 3

Escribe el sistema de ecuaciones en forma de matriz.

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

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

Paso 4

Obtén la forma escalonada reducida por filas.

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Paso 4.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{2}

$\frac{1}{2}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Paso 4.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{2}

$\frac{1}{2}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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

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

Paso 4.1.2

Simplifica

R_{1}

$R_{1}$ .

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

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

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

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

Paso 4.2

Perform the row operation

R_{2} = R_{2} - 3 R_{1}

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Paso 4.2.1

Perform the row operation

R_{2} = R_{2} - 3 R_{1}

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 3 - 3 \cdot 1 & 0 - 3 (\frac{1}{2}) & 0 - 3 \cdot 0 & - 1 - 3 (- \frac{9}{2}) \end{array}]

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 3 - 3 \cdot 1 & 0 - 3 (\frac{1}{2}) & 0 - 3 \cdot 0 & - 1 - 3 (- \frac{9}{2}) \end{array}]$

Paso 4.2.2

Simplifica

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & - \frac{3}{2} & 0 & \frac{25}{2} \end{array}]

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & - \frac{3}{2} & 0 & \frac{25}{2} \end{array}]$

[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & - \frac{3}{2} & 0 & \frac{25}{2} \end{array}]

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & - \frac{3}{2} & 0 & \frac{25}{2} \end{array}]$

Paso 4.3

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{2}{3}

$- \frac{2}{3}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Paso 4.3.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{2}{3}

$- \frac{2}{3}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ - \frac{2}{3} \cdot 0 & - \frac{2}{3} (- \frac{3}{2}) & - \frac{2}{3} \cdot 0 & - \frac{2}{3} \cdot \frac{25}{2} \end{array}]

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ - \frac{2}{3} \cdot 0 & - \frac{2}{3} (- \frac{3}{2}) & - \frac{2}{3} \cdot 0 & - \frac{2}{3} \cdot \frac{25}{2} \end{array}]$

Paso 4.3.2

Simplifica

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & - \frac{9}{2} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

Paso 4.4

Perform the row operation

R_{1} = R_{1} - \frac{1}{2} R_{2}

$R_{1} = R_{1} - \frac{1}{2} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Paso 4.4.1

Perform the row operation

R_{1} = R_{1} - \frac{1}{2} R_{2}

$R_{1} = R_{1} - \frac{1}{2} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 0 - \frac{1}{2} \cdot 0 & - \frac{9}{2} - \frac{1}{2} (- \frac{25}{3}) \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 0 - \frac{1}{2} \cdot 0 & - \frac{9}{2} - \frac{1}{2} (- \frac{25}{3}) \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

Paso 4.4.2

Simplifica

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{3} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]

$[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{3} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{3} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]

$[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{3} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{3} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]

$[\begin{array}{c} 1 & 0 & 0 & - \frac{1}{3} \\ 0 & 1 & 0 & - \frac{25}{3} \end{array}]$

Paso 5

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

x = 0

$x = 0$

y = 0

$y = 0$

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