初级微积分示例

$2 x + y + z = 5$ , $4 x - y - 3 z = 1$ , $8 x + y - z = 5$

解题步骤 1

Write the system as a matrix.

$[\begin{array}{c} 2 & 1 & 1 & 5 \\ 4 & - 1 & - 3 & 1 \\ 8 & 1 & - 1 & 5 \end{array}]$

解题步骤 2

求行简化阶梯形矩阵。

点击获取更多步骤...

解题步骤 2.1

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

点击获取更多步骤...

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 $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ 4 & - 1 & - 3 & 1 \\ 8 & 1 & - 1 & 5 \end{array}]$

解题步骤 2.1.2

化简 $R_{1}$ 。

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ 4 & - 1 & - 3 & 1 \\ 8 & 1 & - 1 & 5 \end{array}]$

解题步骤 2.2

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

点击获取更多步骤...

解题步骤 2.2.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ 4 - 4 \cdot 1 & - 1 - 4 (\frac{1}{2}) & - 3 - 4 (\frac{1}{2}) & 1 - 4 (\frac{5}{2}) \\ 8 & 1 & - 1 & 5 \end{array}]$

解题步骤 2.2.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ 0 & - 3 & - 5 & - 9 \\ 8 & 1 & - 1 & 5 \end{array}]$

解题步骤 2.3

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

点击获取更多步骤...

解题步骤 2.3.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ 0 & - 3 & - 5 & - 9 \\ 8 - 8 \cdot 1 & 1 - 8 (\frac{1}{2}) & - 1 - 8 (\frac{1}{2}) & 5 - 8 (\frac{5}{2}) \end{array}]$

解题步骤 2.3.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ 0 & - 3 & - 5 & - 9 \\ 0 & - 3 & - 5 & - 15 \end{array}]$

解题步骤 2.4

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

点击获取更多步骤...

解题步骤 2.4.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot - 9 \\ 0 & - 3 & - 5 & - 15 \end{array}]$

解题步骤 2.4.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{5}{3} & 3 \\ 0 & - 3 & - 5 & - 15 \end{array}]$

解题步骤 2.5

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

点击获取更多步骤...

解题步骤 2.5.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{5}{3} & 3 \\ 0 + 3 \cdot 0 & - 3 + 3 \cdot 1 & - 5 + 3 (\frac{5}{3}) & - 15 + 3 \cdot 3 \end{array}]$

解题步骤 2.5.2

化简 $R_{3}$ 。

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

解题步骤 2.6

Multiply each element of $R_{3}$ by $- \frac{1}{6}$ to make the entry at $3, 4$ a $1$ .

点击获取更多步骤...

解题步骤 2.6.1

Multiply each element of $R_{3}$ by $- \frac{1}{6}$ to make the entry at $3, 4$ a $1$ .

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

解题步骤 2.6.2

化简 $R_{3}$ 。

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

解题步骤 2.7

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

点击获取更多步骤...

解题步骤 2.7.1

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

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

解题步骤 2.7.2

化简 $R_{2}$ 。

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

解题步骤 2.8

Perform the row operation $R_{1} = R_{1} - \frac{5}{2} R_{3}$ to make the entry at $1, 4$ a $0$ .

点击获取更多步骤...

解题步骤 2.8.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{2} R_{3}$ to make the entry at $1, 4$ a $0$ .

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

解题步骤 2.8.2

化简 $R_{1}$ 。

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

解题步骤 2.9

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

点击获取更多步骤...

解题步骤 2.9.1

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

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

解题步骤 2.9.2

化简 $R_{1}$ 。

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

解题步骤 3

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

$x - \frac{1}{3} z = 0$

$y + \frac{5}{3} z = 0$

$0 = 1$

解题步骤 4

The system is inconsistent so there is no solution.

$无解$

初级微积分 示例

初级微积分示例