有限数学示例

$- 2 d + 11 e + f = 123$ , $4 d + e - 17 f = 217$ , $- d + 19 e + 3 f = 175$

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

从等式两边同时减去 $11 e$ 。

$- 2 d + f = 123 - 11 e$

$4 d + e - 17 f = 217$

$- d + 19 e + 3 f = 175$

解题步骤 1.2

从等式两边同时减去 $e$ 。

$- 2 d + f = 123 - 11 e$

$4 d - 17 f = 217 - e$

$- d + 19 e + 3 f = 175$

解题步骤 1.3

从等式两边同时减去 $19 e$ 。

$- 2 d + f = 123 - 11 e$

$4 d - 17 f = 217 - e$

$- d + 3 f = 175 - 19 e$

$- 2 d + f = 123 - 11 e$

$4 d - 17 f = 217 - e$

$- d + 3 f = 175 - 19 e$

解题步骤 2

Write the system as a matrix.

$[\begin{array}{c} - 2 & 1 & 123 - 11 e \\ 4 & - 17 & 217 - e \\ - 1 & 3 & 175 - 19 e \end{array}]$

解题步骤 3

求行简化阶梯形矩阵。

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解题步骤 3.1

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 3.1.1

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

$[\begin{array}{c} - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 1 & - \frac{1}{2} (123 - 11 e) \\ 4 & - 17 & 217 - e \\ - 1 & 3 & 175 - 19 e \end{array}]$

解题步骤 3.1.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 4 & - 17 & 217 - e \\ - 1 & 3 & 175 - 19 e \end{array}]$

解题步骤 3.2

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

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解题步骤 3.2.1

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

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 4 - 4 \cdot 1 & - 17 - 4 (- \frac{1}{2}) & 217 - e - 4 (- \frac{123}{2} + \frac{11 e}{2}) \\ - 1 & 3 & 175 - 19 e \end{array}]$

解题步骤 3.2.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & - 15 & 463 - 23 e \\ - 1 & 3 & 175 - 19 e \end{array}]$

解题步骤 3.3

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

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解题步骤 3.3.1

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

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & - 15 & 463 - 23 e \\ - 1 + 1 \cdot 1 & 3 - \frac{1}{2} & 175 - 19 e - \frac{123}{2} + \frac{11 e}{2} \end{array}]$

解题步骤 3.3.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & - 15 & 463 - 23 e \\ 0 & \frac{5}{2} & \frac{227 - 27 e}{2} \end{array}]$

解题步骤 3.4

Multiply each element of $R_{2}$ by $- \frac{1}{15}$ to make the entry at $2, 2$ a $1$ .

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解题步骤 3.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{15}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ - \frac{1}{15} \cdot 0 & - \frac{1}{15} \cdot - 15 & - \frac{1}{15} (463 - 23 e) \\ 0 & \frac{5}{2} & \frac{227 - 27 e}{2} \end{array}]$

解题步骤 3.4.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & 1 & - \frac{463}{15} + \frac{23 e}{15} \\ 0 & \frac{5}{2} & \frac{227 - 27 e}{2} \end{array}]$

解题步骤 3.5

Perform the row operation $R_{3} = R_{3} - \frac{5}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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解题步骤 3.5.1

Perform the row operation $R_{3} = R_{3} - \frac{5}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & 1 & - \frac{463}{15} + \frac{23 e}{15} \\ 0 - \frac{5}{2} \cdot 0 & \frac{5}{2} - \frac{5}{2} \cdot 1 & \frac{227 - 27 e}{2} - \frac{5}{2} (- \frac{463}{15} + \frac{23 e}{15}) \end{array}]$

解题步骤 3.5.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & 1 & - \frac{463}{15} + \frac{23 e}{15} \\ 0 & 0 & \frac{572 - 52 e}{3} \end{array}]$

解题步骤 3.6

Multiply each element of $R_{3}$ by $\frac{3}{572 - 52 e}$ to make the entry at $3, 3$ a $1$ .

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解题步骤 3.6.1

Multiply each element of $R_{3}$ by $\frac{3}{572 - 52 e}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & 1 & - \frac{463}{15} + \frac{23 e}{15} \\ \frac{3}{572 - 52 e} \cdot 0 & \frac{3}{572 - 52 e} \cdot 0 & \frac{3}{572 - 52 e} \cdot \frac{572 - 52 e}{3} \end{array}]$

解题步骤 3.6.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & 1 & - \frac{463}{15} + \frac{23 e}{15} \\ 0 & 0 & 1 \end{array}]$

解题步骤 3.7

Perform the row operation $R_{2} = R_{2} - (- \frac{463}{15} + \frac{23 e}{15}) R_{3}$ to make the entry at $2, 3$ a $0$ .

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解题步骤 3.7.1

Perform the row operation $R_{2} = R_{2} - (- \frac{463}{15} + \frac{23 e}{15}) R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 - (- \frac{463}{15} + \frac{23 e}{15}) \cdot 0 & 1 - (- \frac{463}{15} + \frac{23 e}{15}) \cdot 0 & - \frac{463}{15} + \frac{23 e}{15} - (- \frac{463}{15} + \frac{23 e}{15}) \cdot 1 \\ 0 & 0 & 1 \end{array}]$

解题步骤 3.7.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & - \frac{1}{2} & - \frac{123}{2} + \frac{11 e}{2} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 3.8

Perform the row operation $R_{1} = R_{1} - (- \frac{123}{2} + \frac{11 e}{2}) R_{3}$ to make the entry at $1, 3$ a $0$ .

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解题步骤 3.8.1

Perform the row operation $R_{1} = R_{1} - (- \frac{123}{2} + \frac{11 e}{2}) R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - (- \frac{123}{2} + \frac{11 e}{2}) \cdot 0 & - \frac{1}{2} - (- \frac{123}{2} + \frac{11 e}{2}) \cdot 0 & - \frac{123}{2} + \frac{11 e}{2} - (- \frac{123}{2} + \frac{11 e}{2}) \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

解题步骤 3.8.2

化简 $R_{1}$ 。

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

解题步骤 3.9

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

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解题步骤 3.9.1

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

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

解题步骤 3.9.2

化简 $R_{1}$ 。

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

解题步骤 4

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

$d = 0$

$f = 0$

$0 = 1$

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