线性代数示例

$y = - 2 x + 8$ , $y = - 2 x - 4$ , $6 x + 3 y = 6$

解题步骤 1

在等式两边都加上 $2 x$ 。

$y + 2 x = 8, y = - 2 x - 4, 6 x + 3 y = 6$

解题步骤 2

在等式两边都加上 $2 x$ 。

$y + 2 x = 8, y + 2 x = - 4, 6 x + 3 y = 6$

解题步骤 3

以矩阵形式书写方程组。

$[\begin{array}{c} 2 & 1 & 8 \\ 2 & 1 & - 4 \\ 6 & 3 & 6 \end{array}]$

解题步骤 4

求行简化阶梯形矩阵。

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解题步骤 4.1

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 4.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{1}{2} & \frac{8}{2} \\ 2 & 1 & - 4 \\ 6 & 3 & 6 \end{array}]$

解题步骤 4.1.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & \frac{1}{2} & 4 \\ 2 & 1 & - 4 \\ 6 & 3 & 6 \end{array}]$

解题步骤 4.2

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

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解题步骤 4.2.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 4 \\ 2 - 2 \cdot 1 & 1 - 2 (\frac{1}{2}) & - 4 - 2 \cdot 4 \\ 6 & 3 & 6 \end{array}]$

解题步骤 4.2.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & \frac{1}{2} & 4 \\ 0 & 0 & - 12 \\ 6 & 3 & 6 \end{array}]$

解题步骤 4.3

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

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解题步骤 4.3.1

Perform the row operation $R_{3} = R_{3} - 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 4 \\ 0 & 0 & - 12 \\ 6 - 6 \cdot 1 & 3 - 6 (\frac{1}{2}) & 6 - 6 \cdot 4 \end{array}]$

解题步骤 4.3.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & \frac{1}{2} & 4 \\ 0 & 0 & - 12 \\ 0 & 0 & - 18 \end{array}]$

解题步骤 4.4

Multiply each element of $R_{2}$ by $- \frac{1}{12}$ to make the entry at $2, 3$ a $1$ .

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解题步骤 4.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{12}$ to make the entry at $2, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 4 \\ - \frac{1}{12} \cdot 0 & - \frac{1}{12} \cdot 0 & - \frac{1}{12} \cdot - 12 \\ 0 & 0 & - 18 \end{array}]$

解题步骤 4.4.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & \frac{1}{2} & 4 \\ 0 & 0 & 1 \\ 0 & 0 & - 18 \end{array}]$

解题步骤 4.5

Perform the row operation $R_{3} = R_{3} + 18 R_{2}$ to make the entry at $3, 3$ a $0$ .

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解题步骤 4.5.1

Perform the row operation $R_{3} = R_{3} + 18 R_{2}$ to make the entry at $3, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 4 \\ 0 & 0 & 1 \\ 0 + 18 \cdot 0 & 0 + 18 \cdot 0 & - 18 + 18 \cdot 1 \end{array}]$

解题步骤 4.5.2

化简 $R_{3}$ 。

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

解题步骤 4.6

Perform the row operation $R_{1} = R_{1} - 4 R_{2}$ to make the entry at $1, 3$ a $0$ .

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解题步骤 4.6.1

Perform the row operation $R_{1} = R_{1} - 4 R_{2}$ to make the entry at $1, 3$ a $0$ .

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

解题步骤 4.6.2

化简 $R_{1}$ 。

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

解题步骤 5

使用结果矩阵定义方程组的最终解。

$x + \frac{y}{2} = 0$

$0 = 1$

解题步骤 6

从等式两边同时减去 $\frac{y}{2}$ 。

$x = - \frac{y}{2}$

$0 = 1$

解题步骤 7

因为 $0 \neq 1$ ，所以没有解。

无解

线性代数 示例

线性代数示例