Matemática discreta Ejemplos

$x + y = 8$ , $x - y = 4$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 8 \\ 1 & - 1 & 4 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

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

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Paso 2.1.1

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

$[\begin{array}{c} 1 & 1 & 8 \\ 1 - 1 & - 1 - 1 & 4 - 8 \end{array}]$

Paso 2.1.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 8 \\ 0 & - 2 & - 4 \end{array}]$

Paso 2.2

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

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Paso 2.2.1

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

$[\begin{array}{c} 1 & 1 & 8 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 4 \end{array}]$

Paso 2.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 8 \\ 0 & 1 & 2 \end{array}]$

Paso 2.3

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

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Paso 2.3.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 8 - 2 \\ 0 & 1 & 2 \end{array}]$

Paso 2.3.2

Simplifica $R_{1}$ .

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

Paso 3

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

$x = 6$

$y = 2$

Paso 4

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

$(6, 2)$