Matemática discreta Ejemplos

$3 x + 2 y = 14$ , $x - 4 y = - 14$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 3 & 2 & 14 \\ 1 & - 4 & - 14 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{2}{3} & \frac{14}{3} \\ 1 & - 4 & - 14 \end{array}]$

Paso 2.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{14}{3} \\ 1 & - 4 & - 14 \end{array}]$

Paso 2.2

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

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Paso 2.2.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{14}{3} \\ 1 - 1 & - 4 - \frac{2}{3} & - 14 - \frac{14}{3} \end{array}]$

Paso 2.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{14}{3} \\ 0 & - \frac{14}{3} & - \frac{56}{3} \end{array}]$

Paso 2.3

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

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Paso 2.3.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{14}{3} \\ - \frac{3}{14} \cdot 0 & - \frac{3}{14} (- \frac{14}{3}) & - \frac{3}{14} (- \frac{56}{3}) \end{array}]$

Paso 2.3.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{14}{3} \\ 0 & 1 & 4 \end{array}]$

Paso 2.4

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

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Paso 2.4.1

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

$[\begin{array}{c} 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & \frac{14}{3} - \frac{2}{3} \cdot 4 \\ 0 & 1 & 4 \end{array}]$

Paso 2.4.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 2 \\ 0 & 1 & 4 \end{array}]$

Paso 3

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

$x = 2$

$y = 4$

Paso 4

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

$(2, 4)$