유한 수학 예제

$3 x - 2 y = - 12$ , $- x + 3 y = 11$ , $2 x - 4 y = - 16$

단계 1

Write the system as a matrix.

$[\begin{array}{c} 3 & - 2 & - 12 \\ - 1 & 3 & 11 \\ 2 & - 4 & - 16 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{- 12}{3} \\ - 1 & 3 & 11 \\ 2 & - 4 & - 16 \end{array}]$

단계 2.1.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - \frac{2}{3} & - 4 \\ - 1 & 3 & 11 \\ 2 & - 4 & - 16 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & - 4 \\ - 1 + 1 \cdot 1 & 3 - \frac{2}{3} & 11 - 4 \\ 2 & - 4 & - 16 \end{array}]$

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - \frac{2}{3} & - 4 \\ 0 & \frac{7}{3} & 7 \\ 2 & - 4 & - 16 \end{array}]$

단계 2.3

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

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단계 2.3.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & - 4 \\ 0 & \frac{7}{3} & 7 \\ 2 - 2 \cdot 1 & - 4 - 2 (- \frac{2}{3}) & - 16 - 2 \cdot - 4 \end{array}]$

단계 2.3.2

$R_{3}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - \frac{2}{3} & - 4 \\ 0 & \frac{7}{3} & 7 \\ 0 & - \frac{8}{3} & - 8 \end{array}]$

단계 2.4

Multiply each element of $R_{2}$ by $\frac{3}{7}$ to make the entry at $2, 2$ a $1$ .

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단계 2.4.1

Multiply each element of $R_{2}$ by $\frac{3}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & - 4 \\ \frac{3}{7} \cdot 0 & \frac{3}{7} \cdot \frac{7}{3} & \frac{3}{7} \cdot 7 \\ 0 & - \frac{8}{3} & - 8 \end{array}]$

단계 2.4.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - \frac{2}{3} & - 4 \\ 0 & 1 & 3 \\ 0 & - \frac{8}{3} & - 8 \end{array}]$

단계 2.5

Perform the row operation $R_{3} = R_{3} + \frac{8}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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단계 2.5.1

Perform the row operation $R_{3} = R_{3} + \frac{8}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & - 4 \\ 0 & 1 & 3 \\ 0 + \frac{8}{3} \cdot 0 & - \frac{8}{3} + \frac{8}{3} \cdot 1 & - 8 + \frac{8}{3} \cdot 3 \end{array}]$

단계 2.5.2

$R_{3}$ 을 간단히 합니다.

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

단계 2.6

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

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단계 2.6.1

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

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

단계 2.6.2

$R_{1}$ 을 간단히 합니다.

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

단계 3

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

$x = - 2$

$y = 3$

$0 = 0$

단계 4

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

$(- 2, 3)$