유한 수학 예제

$3 x + 6 y - 3 z = 60$ , $x + 2 y + 4 z = - 10$ , $5 x - 2 y + 5 z = - 8$

단계 1

Write the system as a matrix.

$[\begin{array}{c} 3 & 6 & - 3 & 60 \\ 1 & 2 & 4 & - 10 \\ 5 & - 2 & 5 & - 8 \end{array}]$

단계 2

기약 행 사다리꼴을 구합니다.

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단계 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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단계 2.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{6}{3} & \frac{- 3}{3} & \frac{60}{3} \\ 1 & 2 & 4 & - 10 \\ 5 & - 2 & 5 & - 8 \end{array}]$

단계 2.1.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 1 & 2 & 4 & - 10 \\ 5 & - 2 & 5 & - 8 \end{array}]$

단계 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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단계 2.2.1

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

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 1 - 1 & 2 - 2 & 4 + 1 & - 10 - 20 \\ 5 & - 2 & 5 & - 8 \end{array}]$

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 & 0 & 5 & - 30 \\ 5 & - 2 & 5 & - 8 \end{array}]$

단계 2.3

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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단계 2.3.1

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 & 0 & 5 & - 30 \\ 5 - 5 \cdot 1 & - 2 - 5 \cdot 2 & 5 - 5 \cdot - 1 & - 8 - 5 \cdot 20 \end{array}]$

단계 2.3.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 & 0 & 5 & - 30 \\ 0 & - 12 & 10 & - 108 \end{array}]$

단계 2.4

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 & - 12 & 10 & - 108 \\ 0 & 0 & 5 & - 30 \end{array}]$

단계 2.5

Multiply each element of $R_{2}$ by $- \frac{1}{12}$ to make the entry at $2, 2$ a $1$ .

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단계 2.5.1

Multiply each element of $R_{2}$ by $- \frac{1}{12}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ - \frac{1}{12} \cdot 0 & - \frac{1}{12} \cdot - 12 & - \frac{1}{12} \cdot 10 & - \frac{1}{12} \cdot - 108 \\ 0 & 0 & 5 & - 30 \end{array}]$

단계 2.5.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 & 1 & - \frac{5}{6} & 9 \\ 0 & 0 & 5 & - 30 \end{array}]$

단계 2.6

Multiply each element of $R_{3}$ by $\frac{1}{5}$ to make the entry at $3, 3$ a $1$ .

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단계 2.6.1

Multiply each element of $R_{3}$ by $\frac{1}{5}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 & 1 & - \frac{5}{6} & 9 \\ \frac{0}{5} & \frac{0}{5} & \frac{5}{5} & \frac{- 30}{5} \end{array}]$

단계 2.6.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 & 1 & - \frac{5}{6} & 9 \\ 0 & 0 & 1 & - 6 \end{array}]$

단계 2.7

Perform the row operation $R_{2} = R_{2} + \frac{5}{6} R_{3}$ to make the entry at $2, 3$ a $0$ .

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단계 2.7.1

Perform the row operation $R_{2} = R_{2} + \frac{5}{6} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 + \frac{5}{6} \cdot 0 & 1 + \frac{5}{6} \cdot 0 & - \frac{5}{6} + \frac{5}{6} \cdot 1 & 9 + \frac{5}{6} \cdot - 6 \\ 0 & 0 & 1 & - 6 \end{array}]$

단계 2.7.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 2 & - 1 & 20 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & - 6 \end{array}]$

단계 2.8

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

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단계 2.8.1

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

$[\begin{array}{c} 1 + 0 & 2 + 0 & - 1 + 1 \cdot 1 & 20 - 6 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & - 6 \end{array}]$

단계 2.8.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 2 & 0 & 14 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & - 6 \end{array}]$

단계 2.9

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

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단계 2.9.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 14 - 2 \cdot 4 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & - 6 \end{array}]$

단계 2.9.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & 0 & 6 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & - 6 \end{array}]$

단계 3

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

$x = 6$

$y = 4$

$z = - 6$

단계 4

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

$(6, 4, - 6)$