기초 미적분 예제

$- x + 3 y + z = 9$ , $2 x + 5 y = 27$ , $3 x + y - 2 z = - 18$

단계 1

Write the system as a matrix.

$[\begin{array}{c} - 1 & 3 & 1 & 9 \\ 2 & 5 & 0 & 27 \\ 3 & 1 & - 2 & - 18 \end{array}]$

단계 2

기약 행 사다리꼴을 구합니다.

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단계 2.1

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

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단계 2.1.1

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

$[\begin{array}{c} - - 1 & - 1 \cdot 3 & - 1 \cdot 1 & - 1 \cdot 9 \\ 2 & 5 & 0 & 27 \\ 3 & 1 & - 2 & - 18 \end{array}]$

단계 2.1.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 2 & 5 & 0 & 27 \\ 3 & 1 & - 2 & - 18 \end{array}]$

단계 2.2

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

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단계 2.2.1

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

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 2 - 2 \cdot 1 & 5 - 2 \cdot - 3 & 0 - 2 \cdot - 1 & 27 - 2 \cdot - 9 \\ 3 & 1 & - 2 & - 18 \end{array}]$

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 11 & 2 & 45 \\ 3 & 1 & - 2 & - 18 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 11 & 2 & 45 \\ 3 - 3 \cdot 1 & 1 - 3 \cdot - 3 & - 2 - 3 \cdot - 1 & - 18 - 3 \cdot - 9 \end{array}]$

단계 2.3.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 11 & 2 & 45 \\ 0 & 10 & 1 & 9 \end{array}]$

단계 2.4

Multiply each element of $R_{2}$ by $\frac{1}{11}$ to make the entry at $2, 2$ a $1$ .

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단계 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{11}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ \frac{0}{11} & \frac{11}{11} & \frac{2}{11} & \frac{45}{11} \\ 0 & 10 & 1 & 9 \end{array}]$

단계 2.4.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 1 & \frac{2}{11} & \frac{45}{11} \\ 0 & 10 & 1 & 9 \end{array}]$

단계 2.5

Perform the row operation $R_{3} = R_{3} - 10 R_{2}$ to make the entry at $3, 2$ a $0$ .

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단계 2.5.1

Perform the row operation $R_{3} = R_{3} - 10 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 1 & \frac{2}{11} & \frac{45}{11} \\ 0 - 10 \cdot 0 & 10 - 10 \cdot 1 & 1 - 10 (\frac{2}{11}) & 9 - 10 (\frac{45}{11}) \end{array}]$

단계 2.5.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 1 & \frac{2}{11} & \frac{45}{11} \\ 0 & 0 & - \frac{9}{11} & - \frac{351}{11} \end{array}]$

단계 2.6

Multiply each element of $R_{3}$ by $- \frac{11}{9}$ to make the entry at $3, 3$ a $1$ .

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단계 2.6.1

Multiply each element of $R_{3}$ by $- \frac{11}{9}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 1 & \frac{2}{11} & \frac{45}{11} \\ - \frac{11}{9} \cdot 0 & - \frac{11}{9} \cdot 0 & - \frac{11}{9} (- \frac{9}{11}) & - \frac{11}{9} (- \frac{351}{11}) \end{array}]$

단계 2.6.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 1 & \frac{2}{11} & \frac{45}{11} \\ 0 & 0 & 1 & 39 \end{array}]$

단계 2.7

Perform the row operation $R_{2} = R_{2} - \frac{2}{11} R_{3}$ to make the entry at $2, 3$ a $0$ .

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단계 2.7.1

Perform the row operation $R_{2} = R_{2} - \frac{2}{11} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 - \frac{2}{11} \cdot 0 & 1 - \frac{2}{11} \cdot 0 & \frac{2}{11} - \frac{2}{11} \cdot 1 & \frac{45}{11} - \frac{2}{11} \cdot 39 \\ 0 & 0 & 1 & 39 \end{array}]$

단계 2.7.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 3 & - 1 & - 9 \\ 0 & 1 & 0 & - 3 \\ 0 & 0 & 1 & 39 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 + 0 & - 3 + 0 & - 1 + 1 \cdot 1 & - 9 + 1 \cdot 39 \\ 0 & 1 & 0 & - 3 \\ 0 & 0 & 1 & 39 \end{array}]$

단계 2.8.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - 3 & 0 & 30 \\ 0 & 1 & 0 & - 3 \\ 0 & 0 & 1 & 39 \end{array}]$

단계 2.9

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

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단계 2.9.1

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

$[\begin{array}{c} 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 0 + 3 \cdot 0 & 30 + 3 \cdot - 3 \\ 0 & 1 & 0 & - 3 \\ 0 & 0 & 1 & 39 \end{array}]$

단계 2.9.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & 0 & 21 \\ 0 & 1 & 0 & - 3 \\ 0 & 0 & 1 & 39 \end{array}]$

단계 3

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

$x = 21$

$y = - 3$

$z = 39$

단계 4

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

$(21, - 3, 39)$