Matemática discreta Ejemplos

$6 x + 3 y = - 42$ , $x = 15 - 6 y$

Paso 1

Suma $6 y$ a ambos lados de la ecuación.

$6 x + 3 y = - 42$

$x + 6 y = 15$

Paso 2

Write the system as a matrix.

$[\begin{array}{c} 6 & 3 & - 42 \\ 1 & 6 & 15 \end{array}]$

Paso 3

Obtén la forma escalonada reducida por filas.

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Paso 3.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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Paso 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{6}{6} & \frac{3}{6} & \frac{- 42}{6} \\ 1 & 6 & 15 \end{array}]$

Paso 3.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 7 \\ 1 & 6 & 15 \end{array}]$

Paso 3.2

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

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Paso 3.2.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & - 7 \\ 1 - 1 & 6 - \frac{1}{2} & 15 + 7 \end{array}]$

Paso 3.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 7 \\ 0 & \frac{11}{2} & 22 \end{array}]$

Paso 3.3

Multiply each element of $R_{2}$ by $\frac{2}{11}$ to make the entry at $2, 2$ a $1$ .

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Paso 3.3.1

Multiply each element of $R_{2}$ by $\frac{2}{11}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 7 \\ \frac{2}{11} \cdot 0 & \frac{2}{11} \cdot \frac{11}{2} & \frac{2}{11} \cdot 22 \end{array}]$

Paso 3.3.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 7 \\ 0 & 1 & 4 \end{array}]$

Paso 3.4

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

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Paso 3.4.1

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

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & - 7 - \frac{1}{2} \cdot 4 \\ 0 & 1 & 4 \end{array}]$

Paso 3.4.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 9 \\ 0 & 1 & 4 \end{array}]$

Paso 4

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

$x = - 9$

$y = 4$

Paso 5

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

$(- 9, 4)$