有限数学示例

$2 x + 3 y = 3$ , $x + 2 y + 3 z = 1$ , $y + 5 z = 0$

解题步骤 1

Write the system as a matrix.

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

解题步骤 2

求行简化阶梯形矩阵。

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解题步骤 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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解题步骤 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{3}{2} & \frac{0}{2} & \frac{3}{2} \\ 1 & 2 & 3 & 1 \\ 0 & 1 & 5 & 0 \end{array}]$

解题步骤 2.1.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{3}{2} \\ 1 & 2 & 3 & 1 \\ 0 & 1 & 5 & 0 \end{array}]$

解题步骤 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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解题步骤 2.2.1

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

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

解题步骤 2.2.2

化简 $R_{2}$ 。

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

解题步骤 2.3

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

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解题步骤 2.3.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{3}{2} \\ 2 \cdot 0 & 2 (\frac{1}{2}) & 2 \cdot 3 & 2 (- \frac{1}{2}) \\ 0 & 1 & 5 & 0 \end{array}]$

解题步骤 2.3.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{3}{2} \\ 0 & 1 & 6 & - 1 \\ 0 & 1 & 5 & 0 \end{array}]$

解题步骤 2.4

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

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解题步骤 2.4.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{3}{2} \\ 0 & 1 & 6 & - 1 \\ 0 - 0 & 1 - 1 & 5 - 6 & 0 + 1 \end{array}]$

解题步骤 2.4.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{3}{2} \\ 0 & 1 & 6 & - 1 \\ 0 & 0 & - 1 & 1 \end{array}]$

解题步骤 2.5

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

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解题步骤 2.5.1

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

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{3}{2} \\ 0 & 1 & 6 & - 1 \\ - 0 & - 0 & - - 1 & - 1 \cdot 1 \end{array}]$

解题步骤 2.5.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{3}{2} \\ 0 & 1 & 6 & - 1 \\ 0 & 0 & 1 & - 1 \end{array}]$

解题步骤 2.6

Perform the row operation $R_{2} = R_{2} - 6 R_{3}$ to make the entry at $2, 3$ a $0$ .

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解题步骤 2.6.1

Perform the row operation $R_{2} = R_{2} - 6 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{3}{2} & 0 & \frac{3}{2} \\ 0 - 6 \cdot 0 & 1 - 6 \cdot 0 & 6 - 6 \cdot 1 & - 1 - 6 \cdot - 1 \\ 0 & 0 & 1 & - 1 \end{array}]$

解题步骤 2.6.2

化简 $R_{2}$ 。

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

解题步骤 2.7

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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解题步骤 2.7.1

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

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

解题步骤 2.7.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & 0 & 0 & - 6 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & - 1 \end{array}]$

解题步骤 3

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

$x = - 6$

$y = 5$

$z = - 1$

解题步骤 4

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

$(- 6, 5, - 1)$

有限数学 示例

有限数学示例