유한 수학 예제

$x + y = 7$ , $4 x - 3 y + z = 18$

단계 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 0 & 7 \\ 4 & - 3 & 1 & 18 \end{array}]$

단계 2

기약 행 사다리꼴을 구합니다.

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단계 2.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

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단계 2.1.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 0 & 7 \\ 4 - 4 \cdot 1 & - 3 - 4 \cdot 1 & 1 - 4 \cdot 0 & 18 - 4 \cdot 7 \end{array}]$

단계 2.1.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 1 & 0 & 7 \\ 0 & - 7 & 1 & - 10 \end{array}]$

단계 2.2

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

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단계 2.2.1

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

$[\begin{array}{c} 1 & 1 & 0 & 7 \\ - \frac{1}{7} \cdot 0 & - \frac{1}{7} \cdot - 7 & - \frac{1}{7} \cdot 1 & - \frac{1}{7} \cdot - 10 \end{array}]$

단계 2.2.2

$R_{2}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 1 & 0 & 7 \\ 0 & 1 & - \frac{1}{7} & \frac{10}{7} \end{array}]$

단계 2.3

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

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단계 2.3.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 + \frac{1}{7} & 7 - \frac{10}{7} \\ 0 & 1 & - \frac{1}{7} & \frac{10}{7} \end{array}]$

단계 2.3.2

$R_{1}$ 을 간단히 합니다.

$[\begin{array}{c} 1 & 0 & \frac{1}{7} & \frac{39}{7} \\ 0 & 1 & - \frac{1}{7} & \frac{10}{7} \end{array}]$

단계 3

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

$x + \frac{1}{7} z = \frac{39}{7}$

$y - \frac{1}{7} z = \frac{10}{7}$

단계 4

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

$(\frac{39}{7} - \frac{z}{7}, \frac{10}{7} + \frac{z}{7}, z)$