Matemática discreta Exemplos

$2 x + 9 y - 3 z = 0$ , $7 x - 2 y + 4 z = 1$ , $9 x + 7 y + z = 1$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 2 & 9 & - 3 & 0 \\ 7 & - 2 & 4 & 1 \\ 9 & 7 & 1 & 1 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

Toque para ver mais passagens...

Etapa 2.1

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

Toque para ver mais passagens...

Etapa 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{9}{2} & \frac{- 3}{2} & \frac{0}{2} \\ 7 & - 2 & 4 & 1 \\ 9 & 7 & 1 & 1 \end{array}]$

Etapa 2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ 7 & - 2 & 4 & 1 \\ 9 & 7 & 1 & 1 \end{array}]$

Etapa 2.2

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

Toque para ver mais passagens...

Etapa 2.2.1

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

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ 7 - 7 \cdot 1 & - 2 - 7 (\frac{9}{2}) & 4 - 7 (- \frac{3}{2}) & 1 - 7 \cdot 0 \\ 9 & 7 & 1 & 1 \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{67}{2} & \frac{29}{2} & 1 \\ 9 & 7 & 1 & 1 \end{array}]$

Etapa 2.3

Perform the row operation $R_{3} = R_{3} - 9 R_{1}$ to make the entry at $3, 1$ a $0$ .

Toque para ver mais passagens...

Etapa 2.3.1

Perform the row operation $R_{3} = R_{3} - 9 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{67}{2} & \frac{29}{2} & 1 \\ 9 - 9 \cdot 1 & 7 - 9 (\frac{9}{2}) & 1 - 9 (- \frac{3}{2}) & 1 - 9 \cdot 0 \end{array}]$

Etapa 2.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ 0 & - \frac{67}{2} & \frac{29}{2} & 1 \\ 0 & - \frac{67}{2} & \frac{29}{2} & 1 \end{array}]$

Etapa 2.4

Multiply each element of $R_{2}$ by $- \frac{2}{67}$ to make the entry at $2, 2$ a $1$ .

Toque para ver mais passagens...

Etapa 2.4.1

Multiply each element of $R_{2}$ by $- \frac{2}{67}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ - \frac{2}{67} \cdot 0 & - \frac{2}{67} (- \frac{67}{2}) & - \frac{2}{67} \cdot \frac{29}{2} & - \frac{2}{67} \cdot 1 \\ 0 & - \frac{67}{2} & \frac{29}{2} & 1 \end{array}]$

Etapa 2.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - \frac{29}{67} & - \frac{2}{67} \\ 0 & - \frac{67}{2} & \frac{29}{2} & 1 \end{array}]$

Etapa 2.5

Perform the row operation $R_{3} = R_{3} + \frac{67}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

Toque para ver mais passagens...

Etapa 2.5.1

Perform the row operation $R_{3} = R_{3} + \frac{67}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - \frac{29}{67} & - \frac{2}{67} \\ 0 + \frac{67}{2} \cdot 0 & - \frac{67}{2} + \frac{67}{2} \cdot 1 & \frac{29}{2} + \frac{67}{2} (- \frac{29}{67}) & 1 + \frac{67}{2} (- \frac{2}{67}) \end{array}]$

Etapa 2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{9}{2} & - \frac{3}{2} & 0 \\ 0 & 1 & - \frac{29}{67} & - \frac{2}{67} \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 2.6

Perform the row operation $R_{1} = R_{1} - \frac{9}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

Toque para ver mais passagens...

Etapa 2.6.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{2} \cdot 0 & \frac{9}{2} - \frac{9}{2} \cdot 1 & - \frac{3}{2} - \frac{9}{2} (- \frac{29}{67}) & 0 - \frac{9}{2} (- \frac{2}{67}) \\ 0 & 1 & - \frac{29}{67} & - \frac{2}{67} \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 2.6.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{30}{67} & \frac{9}{67} \\ 0 & 1 & - \frac{29}{67} & - \frac{2}{67} \\ 0 & 0 & 0 & 0 \end{array}]$

Etapa 3

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

$x + \frac{30}{67} z = \frac{9}{67}$

$y - \frac{29}{67} z = - \frac{2}{67}$

$0 = 0$

Etapa 4

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

$(\frac{9}{67} - \frac{30 z}{67}, - \frac{2}{67} + \frac{29 z}{67}, z)$