Matemática discreta Ejemplos

$7 x + 4 y = 8$ , $y = - \frac{7}{4} x + 2$

Paso 1

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

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Paso 1.1

Suma $\frac{7}{4} x$ a ambos lados de la ecuación.

$7 x + 4 y = 8$

$y + \frac{7}{4} x = 2$

Paso 1.2

Combina $\frac{7}{4}$ y $x$ .

$7 x + 4 y = 8$

$y + \frac{7 x}{4} = 2$

Paso 1.3

Reordena los términos.

$7 x + 4 y = 8$

$y + \frac{7}{4} x = 2$

Paso 1.4

Reordena $y$ y $\frac{7}{4} x$ .

$7 x + 4 y = 8$

$\frac{7}{4} x + y = 2$

$7 x + 4 y = 8$

$\frac{7}{4} x + y = 2$

Paso 2

Write the system as a matrix.

$[\begin{array}{c} 7 & 4 & 8 \\ \frac{7}{4} & 1 & 2 \end{array}]$

Paso 3

Obtén la forma escalonada reducida por filas.

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Paso 3.1

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

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Paso 3.1.1

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

$[\begin{array}{c} \frac{7}{7} & \frac{4}{7} & \frac{8}{7} \\ \frac{7}{4} & 1 & 2 \end{array}]$

Paso 3.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{4}{7} & \frac{8}{7} \\ \frac{7}{4} & 1 & 2 \end{array}]$

Paso 3.2

Perform the row operation $R_{2} = R_{2} - \frac{7}{4} R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 3.2.1

Perform the row operation $R_{2} = R_{2} - \frac{7}{4} R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{4}{7} & \frac{8}{7} \\ \frac{7}{4} - \frac{7}{4} \cdot 1 & 1 - \frac{7}{4} \cdot \frac{4}{7} & 2 - \frac{7}{4} \cdot \frac{8}{7} \end{array}]$

Paso 3.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{4}{7} & \frac{8}{7} \\ 0 & 0 & 0 \end{array}]$

Paso 4

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

$x + \frac{4}{7} y = \frac{8}{7}$

$0 = 0$

Paso 5

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

$(\frac{8}{7} - \frac{4 y}{7}, y)$