有限数学示例

$x \cdot 1 + x \cdot 2 + x \cdot 3 + s \cdot 1 = 38$ , $- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$ , $x \cdot 3 + s \cdot 3 = 2$ , $- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1

Move variables to the left and constant terms to the right.

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解题步骤 1.1

化简每一项。

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解题步骤 1.1.1

将 $x$ 乘以 $1$ 。

$x + x \cdot 2 + x \cdot 3 + s \cdot 1 = 38$

$- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.1.2

将 $2$ 移到 $x$ 的左侧。

$x + 2 \cdot x + x \cdot 3 + s \cdot 1 = 38$

$- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.1.3

将 $3$ 移到 $x$ 的左侧。

$x + 2 x + 3 \cdot x + s \cdot 1 = 38$

$- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.1.4

将 $s$ 乘以 $1$ 。

$x + 2 x + 3 x + s = 38$

$- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

$x + 2 x + 3 x + s = 38$

$- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.2

将 $x$ 和 $2 x$ 相加。

$3 x + 3 x + s = 38$

$- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.3

将 $3 x$ 和 $3 x$ 相加。

$6 x + s = 38$

$- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.4

将 $6 x$ 和 $s$ 重新排序。

$s + 6 x = 38$

$- x \cdot 1 + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.5

化简每一项。

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解题步骤 1.5.1

将 $- 1$ 乘以 $1$ 。

$s + 6 x = 38$

$- x + x \cdot 2 - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.5.2

将 $2$ 移到 $x$ 的左侧。

$s + 6 x = 38$

$- x + 2 \cdot x - x \cdot 3 + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.5.3

将 $3$ 乘以 $- 1$ 。

$s + 6 x = 38$

$- x + 2 x - 3 x + s \cdot 2 = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.5.4

将 $2$ 移到 $s$ 的左侧。

$s + 6 x = 38$

$- x + 2 x - 3 x + 2 s = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

$s + 6 x = 38$

$- x + 2 x - 3 x + 2 s = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.6

将 $- x$ 和 $2 x$ 相加。

$s + 6 x = 38$

$x - 3 x + 2 s = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.7

从 $x$ 中减去 $3 x$ 。

$s + 6 x = 38$

$- 2 x + 2 s = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.8

将 $- 2 x$ 和 $2 s$ 重新排序。

$s + 6 x = 38$

$2 s - 2 x = 0$

$x \cdot 3 + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.9

化简每一项。

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解题步骤 1.9.1

将 $3$ 移到 $x$ 的左侧。

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 \cdot x + s \cdot 3 = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.9.2

将 $3$ 移到 $s$ 的左侧。

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 x + 3 s = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 x + 3 s = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.10

将 $3 x$ 和 $3 s$ 重新排序。

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 s + 3 x = 2$

$- 200 x \cdot 1 - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.11

化简每一项。

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解题步骤 1.11.1

将 $- 200$ 乘以 $1$ 。

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 s + 3 x = 2$

$- 200 x - 644 x \cdot 2 - 266 x \cdot 3 + z = 0$

解题步骤 1.11.2

将 $2$ 乘以 $- 644$ 。

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 s + 3 x = 2$

$- 200 x - 1288 x - 266 x \cdot 3 + z = 0$

解题步骤 1.11.3

将 $3$ 乘以 $- 266$ 。

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 s + 3 x = 2$

$- 200 x - 1288 x - 798 x + z = 0$

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 s + 3 x = 2$

$- 200 x - 1288 x - 798 x + z = 0$

解题步骤 1.12

从 $- 200 x$ 中减去 $1288 x$ 。

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 s + 3 x = 2$

$- 1488 x - 798 x + z = 0$

解题步骤 1.13

从 $- 1488 x$ 中减去 $798 x$ 。

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 s + 3 x = 2$

$- 2286 x + z = 0$

$s + 6 x = 38$

$2 s - 2 x = 0$

$3 s + 3 x = 2$

$- 2286 x + z = 0$

解题步骤 2

Write the system as a matrix.

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 2 & - 2 & 0 & 0 \\ 3 & 3 & 0 & 2 \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3

求行简化阶梯形矩阵。

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解题步骤 3.1

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 3.1.1

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

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 2 - 2 \cdot 1 & - 2 - 2 \cdot 6 & 0 - 2 \cdot 0 & 0 - 2 \cdot 38 \\ 3 & 3 & 0 & 2 \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3.1.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & - 14 & 0 & - 76 \\ 3 & 3 & 0 & 2 \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3.2

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

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解题步骤 3.2.1

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

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & - 14 & 0 & - 76 \\ 3 - 3 \cdot 1 & 3 - 3 \cdot 6 & 0 - 3 \cdot 0 & 2 - 3 \cdot 38 \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3.2.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & - 14 & 0 & - 76 \\ 0 & - 15 & 0 & - 112 \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3.3

Multiply each element of $R_{2}$ by $- \frac{1}{14}$ to make the entry at $2, 2$ a $1$ .

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解题步骤 3.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{14}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ - \frac{1}{14} \cdot 0 & - \frac{1}{14} \cdot - 14 & - \frac{1}{14} \cdot 0 & - \frac{1}{14} \cdot - 76 \\ 0 & - 15 & 0 & - 112 \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3.3.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 & - 15 & 0 & - 112 \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3.4

Perform the row operation $R_{3} = R_{3} + 15 R_{2}$ to make the entry at $3, 2$ a $0$ .

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解题步骤 3.4.1

Perform the row operation $R_{3} = R_{3} + 15 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 + 15 \cdot 0 & - 15 + 15 \cdot 1 & 0 + 15 \cdot 0 & - 112 + 15 (\frac{38}{7}) \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3.4.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 & 0 & 0 & - \frac{214}{7} \\ 0 & - 2286 & 1 & 0 \end{array}]$

解题步骤 3.5

Perform the row operation $R_{4} = R_{4} + 2286 R_{2}$ to make the entry at $4, 2$ a $0$ .

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解题步骤 3.5.1

Perform the row operation $R_{4} = R_{4} + 2286 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 & 0 & 0 & - \frac{214}{7} \\ 0 + 2286 \cdot 0 & - 2286 + 2286 \cdot 1 & 1 + 2286 \cdot 0 & 0 + 2286 (\frac{38}{7}) \end{array}]$

解题步骤 3.5.2

化简 $R_{4}$ 。

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 & 0 & 0 & - \frac{214}{7} \\ 0 & 0 & 1 & \frac{86868}{7} \end{array}]$

解题步骤 3.6

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 & 0 & 1 & \frac{86868}{7} \\ 0 & 0 & 0 & - \frac{214}{7} \end{array}]$

解题步骤 3.7

Multiply each element of $R_{4}$ by $- \frac{7}{214}$ to make the entry at $4, 4$ a $1$ .

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解题步骤 3.7.1

Multiply each element of $R_{4}$ by $- \frac{7}{214}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 & 0 & 1 & \frac{86868}{7} \\ - \frac{7}{214} \cdot 0 & - \frac{7}{214} \cdot 0 & - \frac{7}{214} \cdot 0 & - \frac{7}{214} (- \frac{214}{7}) \end{array}]$

解题步骤 3.7.2

化简 $R_{4}$ 。

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 & 0 & 1 & \frac{86868}{7} \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 3.8

Perform the row operation $R_{3} = R_{3} - \frac{86868}{7} R_{4}$ to make the entry at $3, 4$ a $0$ .

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解题步骤 3.8.1

Perform the row operation $R_{3} = R_{3} - \frac{86868}{7} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 - \frac{86868}{7} \cdot 0 & 0 - \frac{86868}{7} \cdot 0 & 1 - \frac{86868}{7} \cdot 0 & \frac{86868}{7} - \frac{86868}{7} \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 3.8.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & \frac{38}{7} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 3.9

Perform the row operation $R_{2} = R_{2} - \frac{38}{7} R_{4}$ to make the entry at $2, 4$ a $0$ .

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解题步骤 3.9.1

Perform the row operation $R_{2} = R_{2} - \frac{38}{7} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 - \frac{38}{7} \cdot 0 & 1 - \frac{38}{7} \cdot 0 & 0 - \frac{38}{7} \cdot 0 & \frac{38}{7} - \frac{38}{7} \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 3.9.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 6 & 0 & 38 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 3.10

Perform the row operation $R_{1} = R_{1} - 38 R_{4}$ to make the entry at $1, 4$ a $0$ .

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解题步骤 3.10.1

Perform the row operation $R_{1} = R_{1} - 38 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 38 \cdot 0 & 6 - 38 \cdot 0 & 0 - 38 \cdot 0 & 38 - 38 \cdot 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

解题步骤 3.10.2

化简 $R_{1}$ 。

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

解题步骤 3.11

Perform the row operation $R_{1} = R_{1} - 6 R_{2}$ to make the entry at $1, 2$ a $0$ .

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解题步骤 3.11.1

Perform the row operation $R_{1} = R_{1} - 6 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

解题步骤 3.11.2

化简 $R_{1}$ 。

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

解题步骤 4

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

$s = 0$

$x = 0$

$z = 0$

$0 = 1$

解题步骤 5

The system is inconsistent so there is no solution.

$无解$

有限数学 示例

有限数学示例