Álgebra linear Exemplos

$2 x + y - z = 5$ , $x + 4 y + 2 z = 16$ , $15 x + 6 y - 2 z = 12$

Etapa 1

Write the system as a matrix.

$[\begin{array}{c} 2 & 1 & - 1 & 5 \\ 1 & 4 & 2 & 16 \\ 15 & 6 & - 2 & 12 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

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

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Etapa 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{1}{2} & \frac{- 1}{2} & \frac{5}{2} \\ 1 & 4 & 2 & 16 \\ 15 & 6 & - 2 & 12 \end{array}]$

Etapa 2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 1 & 4 & 2 & 16 \\ 15 & 6 & - 2 & 12 \end{array}]$

Etapa 2.2

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

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Etapa 2.2.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 1 - 1 & 4 - \frac{1}{2} & 2 + \frac{1}{2} & 16 - \frac{5}{2} \\ 15 & 6 & - 2 & 12 \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & \frac{7}{2} & \frac{5}{2} & \frac{27}{2} \\ 15 & 6 & - 2 & 12 \end{array}]$

Etapa 2.3

Perform the row operation $R_{3} = R_{3} - 15 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 2.3.1

Perform the row operation $R_{3} = R_{3} - 15 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & \frac{7}{2} & \frac{5}{2} & \frac{27}{2} \\ 15 - 15 \cdot 1 & 6 - 15 (\frac{1}{2}) & - 2 - 15 (- \frac{1}{2}) & 12 - 15 (\frac{5}{2}) \end{array}]$

Etapa 2.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & \frac{7}{2} & \frac{5}{2} & \frac{27}{2} \\ 0 & - \frac{3}{2} & \frac{11}{2} & - \frac{51}{2} \end{array}]$

Etapa 2.4

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

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Etapa 2.4.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ \frac{2}{7} \cdot 0 & \frac{2}{7} \cdot \frac{7}{2} & \frac{2}{7} \cdot \frac{5}{2} & \frac{2}{7} \cdot \frac{27}{2} \\ 0 & - \frac{3}{2} & \frac{11}{2} & - \frac{51}{2} \end{array}]$

Etapa 2.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{5}{7} & \frac{27}{7} \\ 0 & - \frac{3}{2} & \frac{11}{2} & - \frac{51}{2} \end{array}]$

Etapa 2.5

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

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Etapa 2.5.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{5}{7} & \frac{27}{7} \\ 0 + \frac{3}{2} \cdot 0 & - \frac{3}{2} + \frac{3}{2} \cdot 1 & \frac{11}{2} + \frac{3}{2} \cdot \frac{5}{7} & - \frac{51}{2} + \frac{3}{2} \cdot \frac{27}{7} \end{array}]$

Etapa 2.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{5}{7} & \frac{27}{7} \\ 0 & 0 & \frac{46}{7} & - \frac{138}{7} \end{array}]$

Etapa 2.6

Multiply each element of $R_{3}$ by $\frac{7}{46}$ to make the entry at $3, 3$ a $1$ .

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Etapa 2.6.1

Multiply each element of $R_{3}$ by $\frac{7}{46}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{5}{7} & \frac{27}{7} \\ \frac{7}{46} \cdot 0 & \frac{7}{46} \cdot 0 & \frac{7}{46} \cdot \frac{46}{7} & \frac{7}{46} (- \frac{138}{7}) \end{array}]$

Etapa 2.6.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{5}{7} & \frac{27}{7} \\ 0 & 0 & 1 & - 3 \end{array}]$

Etapa 2.7

Perform the row operation $R_{2} = R_{2} - \frac{5}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 2.7.1

Perform the row operation $R_{2} = R_{2} - \frac{5}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 - \frac{5}{7} \cdot 0 & 1 - \frac{5}{7} \cdot 0 & \frac{5}{7} - \frac{5}{7} \cdot 1 & \frac{27}{7} - \frac{5}{7} \cdot - 3 \\ 0 & 0 & 1 & - 3 \end{array}]$

Etapa 2.7.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 1 & - 3 \end{array}]$

Etapa 2.8

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

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Etapa 2.8.1

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

$[\begin{array}{c} 1 + \frac{1}{2} \cdot 0 & \frac{1}{2} + \frac{1}{2} \cdot 0 & - \frac{1}{2} + \frac{1}{2} \cdot 1 & \frac{5}{2} + \frac{1}{2} \cdot - 3 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 1 & - 3 \end{array}]$

Etapa 2.8.2

Simplifique $R_{1}$ .

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

Etapa 2.9

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

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Etapa 2.9.1

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

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

Etapa 2.9.2

Simplifique $R_{1}$ .

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

Etapa 3

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

$x = - 2$

$y = 6$

$z = - 3$

Etapa 4

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

$(- 2, 6, - 3)$