Matemática discreta Exemplos

$3 u - 2 v = 12$ , $7 u + 2 v = 8$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 3 & - 2 & 12 \\ 7 & 2 & 8 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

Multiply each element of $R_{1}$ by $\frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

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Etapa 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{12}{3} \\ 7 & 2 & 8 \end{array}]$

Etapa 2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 4 \\ 7 & 2 & 8 \end{array}]$

Etapa 2.2

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

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Etapa 2.2.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & 4 \\ 7 - 7 \cdot 1 & 2 - 7 (- \frac{2}{3}) & 8 - 7 \cdot 4 \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 4 \\ 0 & \frac{20}{3} & - 20 \end{array}]$

Etapa 2.3

Multiply each element of $R_{2}$ by $\frac{3}{20}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.3.1

Multiply each element of $R_{2}$ by $\frac{3}{20}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 4 \\ \frac{3}{20} \cdot 0 & \frac{3}{20} \cdot \frac{20}{3} & \frac{3}{20} \cdot - 20 \end{array}]$

Etapa 2.3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 4 \\ 0 & 1 & - 3 \end{array}]$

Etapa 2.4

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

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Etapa 2.4.1

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

$[\begin{array}{c} 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & 4 + \frac{2}{3} \cdot - 3 \\ 0 & 1 & - 3 \end{array}]$

Etapa 2.4.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 2 \\ 0 & 1 & - 3 \end{array}]$

Etapa 3

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

$u = 2$

$v = - 3$

Etapa 4

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

$(2, - 3)$