Matemática discreta Ejemplos

$x + 3 y + z = 84$ , $7 y + 2 z = 42$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 3 & 1 & 84 \\ 0 & 7 & 2 & 42 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

Multiply each element of $R_{2}$ by $\frac{1}{7}$ to make the entry at $2, 2$ a $1$ .

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Paso 2.1.1

Multiply each element of $R_{2}$ by $\frac{1}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 3 & 1 & 84 \\ \frac{0}{7} & \frac{7}{7} & \frac{2}{7} & \frac{42}{7} \end{array}]$

Paso 2.1.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 3 & 1 & 84 \\ 0 & 1 & \frac{2}{7} & 6 \end{array}]$

Paso 2.2

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

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Paso 2.2.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 3 - 3 \cdot 1 & 1 - 3 (\frac{2}{7}) & 84 - 3 \cdot 6 \\ 0 & 1 & \frac{2}{7} & 6 \end{array}]$

Paso 2.2.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{7} & 66 \\ 0 & 1 & \frac{2}{7} & 6 \end{array}]$

Paso 3

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

$x + \frac{1}{7} z = 66$

$y + \frac{2}{7} z = 6$

Paso 4

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

$(66 - \frac{z}{7}, 6 - \frac{2 z}{7}, z)$