유한 수학 예제

$2 x - 3 y = - 8$ , $5 x + 3 y = 1$

단계 1

Write the system as a matrix.

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

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 $1$ .

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

단계 2.1.2

$R_{1}$ 을 간단히 합니다.

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

단계 2.2

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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단계 2.2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & - 4 \\ 5 - 5 \cdot 1 & 3 - 5 (- \frac{3}{2}) & 1 - 5 \cdot - 4 \end{array}]$

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & - \frac{3}{2} & - 4 \\ 0 & \frac{21}{2} & 21 \end{array}]$

단계 2.3

Multiply each element of $R_{2}$ by $\frac{2}{21}$ to make the entry at $2, 2$ a $1$ .

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단계 2.3.1

Multiply each element of $R_{2}$ by $\frac{2}{21}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & - 4 \\ \frac{2}{21} \cdot 0 & \frac{2}{21} \cdot \frac{21}{2} & \frac{2}{21} \cdot 21 \end{array}]$

단계 2.3.2

$R_{2}$ 을 간단히 합니다.

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

단계 2.4

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

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단계 2.4.1

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

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

단계 2.4.2

$R_{1}$ 을 간단히 합니다.

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

단계 3

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

$x = - 1$

$y = 2$

단계 4

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

$(- 1, 2)$