Matemática discreta Ejemplos

$9 x - 7 y = 63$ , $45 x - 35 = 18$

Paso 1

Mueve todos los términos que no contengan una variable al lado derecho de la ecuación.

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Paso 1.1

Suma $35$ a ambos lados de la ecuación.

$9 x - 7 y = 63$

$45 x = 18 + 35$

Paso 1.2

Suma $18$ y $35$ .

$9 x - 7 y = 63$

$45 x = 53$

$9 x - 7 y = 63$

$45 x = 53$

Paso 2

Write the system as a matrix.

$[\begin{array}{c} 9 & - 7 & 63 \\ 45 & 0 & 53 \end{array}]$

Paso 3

Obtén la forma escalonada reducida por filas.

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Paso 3.1

Multiply each element of $R_{1}$ by $\frac{1}{9}$ to make the entry at $1, 1$ a $1$ .

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Paso 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{9}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{9}{9} & \frac{- 7}{9} & \frac{63}{9} \\ 45 & 0 & 53 \end{array}]$

Paso 3.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{7}{9} & 7 \\ 45 & 0 & 53 \end{array}]$

Paso 3.2

Perform the row operation $R_{2} = R_{2} - 45 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 3.2.1

Perform the row operation $R_{2} = R_{2} - 45 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{7}{9} & 7 \\ 45 - 45 \cdot 1 & 0 - 45 (- \frac{7}{9}) & 53 - 45 \cdot 7 \end{array}]$

Paso 3.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{7}{9} & 7 \\ 0 & 35 & - 262 \end{array}]$

Paso 3.3

Multiply each element of $R_{2}$ by $\frac{1}{35}$ to make the entry at $2, 2$ a $1$ .

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Paso 3.3.1

Multiply each element of $R_{2}$ by $\frac{1}{35}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{7}{9} & 7 \\ \frac{0}{35} & \frac{35}{35} & \frac{- 262}{35} \end{array}]$

Paso 3.3.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{7}{9} & 7 \\ 0 & 1 & - \frac{262}{35} \end{array}]$

Paso 3.4

Perform the row operation $R_{1} = R_{1} + \frac{7}{9} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Paso 3.4.1

Perform the row operation $R_{1} = R_{1} + \frac{7}{9} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{7}{9} \cdot 0 & - \frac{7}{9} + \frac{7}{9} \cdot 1 & 7 + \frac{7}{9} (- \frac{262}{35}) \\ 0 & 1 & - \frac{262}{35} \end{array}]$

Paso 3.4.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{53}{45} \\ 0 & 1 & - \frac{262}{35} \end{array}]$

Paso 4

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

$x = \frac{53}{45}$

$y = - \frac{262}{35}$

Paso 5

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

$(\frac{53}{45}, - \frac{262}{35})$