Matemática discreta Exemplos

$x - 5 y = 35$ , $5 x - 4 y = 28$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 5 & 35 \\ 5 & - 4 & 28 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 2.1.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 5 & 35 \\ 5 - 5 \cdot 1 & - 4 - 5 \cdot - 5 & 28 - 5 \cdot 35 \end{array}]$

Etapa 2.1.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 5 & 35 \\ 0 & 21 & - 147 \end{array}]$

Etapa 2.2

Multiply each element of $R_{2}$ by $\frac{1}{21}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.2.1

Multiply each element of $R_{2}$ by $\frac{1}{21}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 5 & 35 \\ \frac{0}{21} & \frac{21}{21} & \frac{- 147}{21} \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - 5 & 35 \\ 0 & 1 & - 7 \end{array}]$

Etapa 2.3

Perform the row operation $R_{1} = R_{1} + 5 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 2.3.1

Perform the row operation $R_{1} = R_{1} + 5 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 5 \cdot 0 & - 5 + 5 \cdot 1 & 35 + 5 \cdot - 7 \\ 0 & 1 & - 7 \end{array}]$

Etapa 2.3.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$x = 0$

$y = - 7$

Etapa 4

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

$(0, - 7)$