유한 수학 예제

$- 2 x - y - 9 z = - 77$ , $- 7 x + 4 y + 7 z = 30$ , $- 3 x - 9 y + z = - 44$

단계 1

Write the system as a matrix.

$[\begin{array}{c} - 2 & - 1 & - 9 & - 77 \\ - 7 & 4 & 7 & 30 \\ - 3 & - 9 & 1 & - 44 \end{array}]$

단계 2

기약 행 사다리꼴을 구합니다.

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단계 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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단계 2.1.1

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

$[\begin{array}{c} - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 1 & - \frac{1}{2} \cdot - 9 & - \frac{1}{2} \cdot - 77 \\ - 7 & 4 & 7 & 30 \\ - 3 & - 9 & 1 & - 44 \end{array}]$

단계 2.1.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ - 7 & 4 & 7 & 30 \\ - 3 & - 9 & 1 & - 44 \end{array}]$

단계 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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단계 2.2.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ - 7 + 7 \cdot 1 & 4 + 7 (\frac{1}{2}) & 7 + 7 (\frac{9}{2}) & 30 + 7 (\frac{77}{2}) \\ - 3 & - 9 & 1 & - 44 \end{array}]$

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & \frac{15}{2} & \frac{77}{2} & \frac{599}{2} \\ - 3 & - 9 & 1 & - 44 \end{array}]$

단계 2.3

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

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단계 2.3.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & \frac{15}{2} & \frac{77}{2} & \frac{599}{2} \\ - 3 + 3 \cdot 1 & - 9 + 3 (\frac{1}{2}) & 1 + 3 (\frac{9}{2}) & - 44 + 3 (\frac{77}{2}) \end{array}]$

단계 2.3.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & \frac{15}{2} & \frac{77}{2} & \frac{599}{2} \\ 0 & - \frac{15}{2} & \frac{29}{2} & \frac{143}{2} \end{array}]$

단계 2.4

Multiply each element of $R_{2}$ by $\frac{2}{15}$ to make the entry at $2, 2$ a $1$ .

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단계 2.4.1

Multiply each element of $R_{2}$ by $\frac{2}{15}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ \frac{2}{15} \cdot 0 & \frac{2}{15} \cdot \frac{15}{2} & \frac{2}{15} \cdot \frac{77}{2} & \frac{2}{15} \cdot \frac{599}{2} \\ 0 & - \frac{15}{2} & \frac{29}{2} & \frac{143}{2} \end{array}]$

단계 2.4.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & 1 & \frac{77}{15} & \frac{599}{15} \\ 0 & - \frac{15}{2} & \frac{29}{2} & \frac{143}{2} \end{array}]$

단계 2.5

Perform the row operation $R_{3} = R_{3} + \frac{15}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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단계 2.5.1

Perform the row operation $R_{3} = R_{3} + \frac{15}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & 1 & \frac{77}{15} & \frac{599}{15} \\ 0 + \frac{15}{2} \cdot 0 & - \frac{15}{2} + \frac{15}{2} \cdot 1 & \frac{29}{2} + \frac{15}{2} \cdot \frac{77}{15} & \frac{143}{2} + \frac{15}{2} \cdot \frac{599}{15} \end{array}]$

단계 2.5.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & 1 & \frac{77}{15} & \frac{599}{15} \\ 0 & 0 & 53 & 371 \end{array}]$

단계 2.6

Multiply each element of $R_{3}$ by $\frac{1}{53}$ to make the entry at $3, 3$ a $1$ .

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단계 2.6.1

Multiply each element of $R_{3}$ by $\frac{1}{53}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & 1 & \frac{77}{15} & \frac{599}{15} \\ \frac{0}{53} & \frac{0}{53} & \frac{53}{53} & \frac{371}{53} \end{array}]$

단계 2.6.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & 1 & \frac{77}{15} & \frac{599}{15} \\ 0 & 0 & 1 & 7 \end{array}]$

단계 2.7

Perform the row operation $R_{2} = R_{2} - \frac{77}{15} R_{3}$ to make the entry at $2, 3$ a $0$ .

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단계 2.7.1

Perform the row operation $R_{2} = R_{2} - \frac{77}{15} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 - \frac{77}{15} \cdot 0 & 1 - \frac{77}{15} \cdot 0 & \frac{77}{15} - \frac{77}{15} \cdot 1 & \frac{599}{15} - \frac{77}{15} \cdot 7 \\ 0 & 0 & 1 & 7 \end{array}]$

단계 2.7.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{9}{2} & \frac{77}{2} \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 7 \end{array}]$

단계 2.8

Perform the row operation $R_{1} = R_{1} - \frac{9}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

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단계 2.8.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{2} \cdot 0 & \frac{1}{2} - \frac{9}{2} \cdot 0 & \frac{9}{2} - \frac{9}{2} \cdot 1 & \frac{77}{2} - \frac{9}{2} \cdot 7 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 7 \end{array}]$

단계 2.8.2

$R_{1}$ 을 간단히 합니다.

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

단계 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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단계 2.9.1

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

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

단계 2.9.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 7 \end{array}]$

단계 3

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

$x = 5$

$y = 4$

$z = 7$

단계 4

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

$(5, 4, 7)$