Matemática discreta Ejemplos

$x \cdot 1 + 9 x \cdot 2 = 67$ , $4 x \cdot 1 - x \cdot 2 = 9$

Paso 1

Move variables to the left and constant terms to the right.

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Paso 1.1

Simplifica cada término.

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Paso 1.1.1

Multiplica $x$ por $1$ .

$x + 9 x \cdot 2 = 67$

$4 x \cdot 1 - x \cdot 2 = 9$

Paso 1.1.2

Multiplica $2$ por $9$ .

$x + 18 x = 67$

$4 x \cdot 1 - x \cdot 2 = 9$

$x + 18 x = 67$

$4 x \cdot 1 - x \cdot 2 = 9$

Paso 1.2

Suma $x$ y $18 x$ .

$19 x = 67$

$4 x \cdot 1 - x \cdot 2 = 9$

Paso 1.3

Simplifica cada término.

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Paso 1.3.1

Multiplica $4$ por $1$ .

$19 x = 67$

$4 x - x \cdot 2 = 9$

Paso 1.3.2

Multiplica $2$ por $- 1$ .

$19 x = 67$

$4 x - 2 x = 9$

$19 x = 67$

$4 x - 2 x = 9$

Paso 1.4

Resta $2 x$ de $4 x$ .

$19 x = 67$

$2 x = 9$

$19 x = 67$

$2 x = 9$

Paso 2

Write the system as a matrix.

$[\begin{array}{c} 19 & 67 \\ 2 & 9 \end{array}]$

Paso 3

Obtén la forma escalonada reducida por filas.

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Paso 3.1

Multiply each element of $R_{1}$ by $\frac{1}{19}$ to make the entry at $1, 1$ a $1$ .

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Paso 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{19}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{19}{19} & \frac{67}{19} \\ 2 & 9 \end{array}]$

Paso 3.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{67}{19} \\ 2 & 9 \end{array}]$

Paso 3.2

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 3.2.1

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

$[\begin{array}{c} 1 & \frac{67}{19} \\ 2 - 2 \cdot 1 & 9 - 2 (\frac{67}{19}) \end{array}]$

Paso 3.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{67}{19} \\ 0 & \frac{37}{19} \end{array}]$

Paso 3.3

Multiply each element of $R_{2}$ by $\frac{19}{37}$ to make the entry at $2, 2$ a $1$ .

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Paso 3.3.1

Multiply each element of $R_{2}$ by $\frac{19}{37}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{67}{19} \\ \frac{19}{37} \cdot 0 & \frac{19}{37} \cdot \frac{37}{19} \end{array}]$

Paso 3.3.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{67}{19} \\ 0 & 1 \end{array}]$

Paso 3.4

Perform the row operation $R_{1} = R_{1} - \frac{67}{19} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Paso 3.4.1

Perform the row operation $R_{1} = R_{1} - \frac{67}{19} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{67}{19} \cdot 0 & \frac{67}{19} - \frac{67}{19} \cdot 1 \\ 0 & 1 \end{array}]$

Paso 3.4.2

Simplifica $R_{1}$ .

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

Paso 4

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

$x = 0$

$0 = 1$

Paso 5

The system is inconsistent so there is no solution.

$No hay solución$