Matemática discreta Ejemplos

$3 x - y = 8$ , $- 9 x + 3 y = - 10$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 3 & - 1 & 8 \\ - 9 & 3 & - 10 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

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

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Paso 2.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 1}{3} & \frac{8}{3} \\ - 9 & 3 & - 10 \end{array}]$

Paso 2.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{8}{3} \\ - 9 & 3 & - 10 \end{array}]$

Paso 2.2

Perform the row operation $R_{2} = R_{2} + 9 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 2.2.1

Perform the row operation $R_{2} = R_{2} + 9 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{8}{3} \\ - 9 + 9 \cdot 1 & 3 + 9 (- \frac{1}{3}) & - 10 + 9 (\frac{8}{3}) \end{array}]$

Paso 2.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{8}{3} \\ 0 & 0 & 14 \end{array}]$

Paso 2.3

Multiply each element of $R_{2}$ by $\frac{1}{14}$ to make the entry at $2, 3$ a $1$ .

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Paso 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{14}$ to make the entry at $2, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{8}{3} \\ \frac{0}{14} & \frac{0}{14} & \frac{14}{14} \end{array}]$

Paso 2.3.2

Simplifica $R_{2}$ .

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

Paso 2.4

Perform the row operation $R_{1} = R_{1} - \frac{8}{3} R_{2}$ to make the entry at $1, 3$ a $0$ .

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Paso 2.4.1

Perform the row operation $R_{1} = R_{1} - \frac{8}{3} R_{2}$ to make the entry at $1, 3$ a $0$ .

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

Paso 2.4.2

Simplifica $R_{1}$ .

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

Paso 3

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

$x - \frac{1}{3} y = 0$

$0 = 1$

Paso 4

The system is inconsistent so there is no solution.

$No hay solución$