유한 수학 예제

$3 x + y = 1$ , $- 7 x - 2 y = - 1$

단계 1

Write the system as a matrix.

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

단계 2

기약 행 사다리꼴을 구합니다.

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단계 2.1

Multiply each element of $R_{1}$ by $\frac{1}{3}$ 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 $1$ .

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

단계 2.1.2

$R_{1}$ 을 간단히 합니다.

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

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 $0$ .

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

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

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

단계 2.3

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

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단계 2.3.1

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

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

단계 2.3.2

$R_{2}$ 을 간단히 합니다.

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

단계 2.4

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

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단계 2.4.1

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

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

단계 2.4.2

$R_{1}$ 을 간단히 합니다.

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

단계 3

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

$x = - 1$

$y = 4$

단계 4

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

$(- 1, 4)$