线性代数示例

$9 x + 9 x - 7 x = 6$ ,

$- 7 x + 0 - 1 x = - 10$ ,

$9 x + 6 x + 8 x = 45$

解题步骤 1

化简。

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解题步骤 1.1

将

$9 x$ 和

$9 x$ 相加。

$18 x - 7 x = 6, - 7 x + 0 - 1 x = - 10, 9 x + 6 x + 8 x = 45$

解题步骤 1.2

从

$18 x$ 中减去

$7 x$ 。

$11 x = 6, - 7 x + 0 - 1 x = - 10, 9 x + 6 x + 8 x = 45$

解题步骤 2

化简。

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解题步骤 2.1

将

$- 7 x$ 和

$0$ 相加。

$11 x = 6, - 7 x - 1 x = - 10, 9 x + 6 x + 8 x = 45$

解题步骤 2.2

将

$- 1 x$ 重写为

$- x$ 。

$11 x = 6, - 7 x - x = - 10, 9 x + 6 x + 8 x = 45$

解题步骤 2.3

从

$- 7 x$ 中减去

$x$ 。

$11 x = 6, - 8 x = - 10, 9 x + 6 x + 8 x = 45$

解题步骤 3

化简。

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解题步骤 3.1

将

$9 x$ 和

$6 x$ 相加。

$11 x = 6, - 8 x = - 10, 15 x + 8 x = 45$

解题步骤 3.2

将

$15 x$ 和

$8 x$ 相加。

$11 x = 6, - 8 x = - 10, 23 x = 45$

解题步骤 4

以矩阵形式书写方程组。

$[\begin{array}{c} 11 & 6 \\ - 8 & - 10 \\ 23 & 45 \end{array}]$

解题步骤 5

求行简化阶梯形矩阵。

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解题步骤 5.1

Multiply each element of

$R_{1}$ by

$\frac{1}{11}$ to make the entry at

$1, 1$ a

$1$ .

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解题步骤 5.1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{11}$ to make the entry at

$1, 1$ a

$1$ .

$[\begin{array}{c} \frac{11}{11} & \frac{6}{11} \\ - 8 & - 10 \\ 23 & 45 \end{array}]$

解题步骤 5.1.2

化简

$R_{1}$ 。

$[\begin{array}{c} 1 & \frac{6}{11} \\ - 8 & - 10 \\ 23 & 45 \end{array}]$

解题步骤 5.2

Perform the row operation

$R_{2} = R_{2} + 8 R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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解题步骤 5.2.1

Perform the row operation

$R_{2} = R_{2} + 8 R_{1}$ to make the entry at

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{6}{11} \\ - 8 + 8 \cdot 1 & - 10 + 8 (\frac{6}{11}) \\ 23 & 45 \end{array}]$

解题步骤 5.2.2

化简

$R_{2}$ 。

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 23 & 45 \end{array}]$

解题步骤 5.3

Perform the row operation

$R_{3} = R_{3} - 23 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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解题步骤 5.3.1

Perform the row operation

$R_{3} = R_{3} - 23 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 23 - 23 \cdot 1 & 45 - 23 (\frac{6}{11}) \end{array}]$

解题步骤 5.3.2

化简

$R_{3}$ 。

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 0 & \frac{357}{11} \end{array}]$

解题步骤 5.4

Multiply each element of

$R_{2}$ by

$- \frac{11}{62}$ to make the entry at

$2, 2$ a

$1$ .

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解题步骤 5.4.1

Multiply each element of

$R_{2}$ by

$- \frac{11}{62}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & \frac{6}{11} \\ - \frac{11}{62} \cdot 0 & - \frac{11}{62} (- \frac{62}{11}) \\ 0 & \frac{357}{11} \end{array}]$

解题步骤 5.4.2

化简

$R_{2}$ 。

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & \frac{357}{11} \end{array}]$

解题步骤 5.5

Perform the row operation

$R_{3} = R_{3} - \frac{357}{11} R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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解题步骤 5.5.1

Perform the row operation

$R_{3} = R_{3} - \frac{357}{11} R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 - \frac{357}{11} \cdot 0 & \frac{357}{11} - \frac{357}{11} \cdot 1 \end{array}]$

解题步骤 5.5.2

化简

$R_{3}$ 。

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & 0 \end{array}]$

解题步骤 5.6

Perform the row operation

$R_{1} = R_{1} - \frac{6}{11} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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解题步骤 5.6.1

Perform the row operation

$R_{1} = R_{1} - \frac{6}{11} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - \frac{6}{11} \cdot 0 & \frac{6}{11} - \frac{6}{11} \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

解题步骤 5.6.2

化简

$R_{1}$ 。

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

解题步骤 6

使用结果矩阵定义方程组的最终解。

$x = 0$

$0 = 1$

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