有限数学示例

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

解题步骤 1

Write the system as a matrix.

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

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

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

点击获取更多步骤...

解题步骤 2.1.1

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

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

解题步骤 2.1.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & - 4 & 4 & 20 \\ 1 & - 2 & 4 & 4 \end{array}]$

解题步骤 2.2

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

点击获取更多步骤...

解题步骤 2.2.1

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

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & - 4 & 4 & 20 \\ 1 - 1 & - 2 - 1 & 4 + 1 & 4 + 5 \end{array}]$

解题步骤 2.2.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ 0 & - 4 & 4 & 20 \\ 0 & - 3 & 5 & 9 \end{array}]$

解题步骤 2.3

Multiply each element of $R_{2}$ by $- \frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

点击获取更多步骤...

解题步骤 2.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 1 & - 5 \\ - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot - 4 & - \frac{1}{4} \cdot 4 & - \frac{1}{4} \cdot 20 \\ 0 & - 3 & 5 & 9 \end{array}]$

解题步骤 2.3.2

化简 $R_{2}$ 。

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

解题步骤 2.4

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 2.4.1

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

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

解题步骤 2.4.2

化简 $R_{3}$ 。

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

解题步骤 2.5

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

点击获取更多步骤...

解题步骤 2.5.1

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

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

解题步骤 2.5.2

化简 $R_{3}$ 。

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

解题步骤 2.6

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

点击获取更多步骤...

解题步骤 2.6.1

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

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

解题步骤 2.6.2

化简 $R_{2}$ 。

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

解题步骤 2.7

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

点击获取更多步骤...

解题步骤 2.7.1

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

$[\begin{array}{c} 1 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & - 5 - 3 \\ 0 & 1 & 0 & - 8 \\ 0 & 0 & 1 & - 3 \end{array}]$

解题步骤 2.7.2

化简 $R_{1}$ 。

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

解题步骤 2.8

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

点击获取更多步骤...

解题步骤 2.8.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & - 8 + 8 \\ 0 & 1 & 0 & - 8 \\ 0 & 0 & 1 & - 3 \end{array}]$

解题步骤 2.8.2

化简 $R_{1}$ 。

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

解题步骤 3

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

$x = 0$

$y = - 8$

$z = - 3$

解题步骤 4

The solution is the set of ordered pairs that make the system true.

$(0, - 8, - 3)$

有限数学 示例

有限数学示例