有限数学示例

$x - 2 y = 0$ , $x + y + z = 310$ , $45000 x + 40000 y + 80000 z = 16000000$

解题步骤 1

以矩阵形式表示方程组。

$[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$

解题步骤 2

Find the determinant of the coefficient matrix $[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}]$ .

点击获取更多步骤...

解题步骤 2.1

Write $[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}]$ in determinant notation.

$| \begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array} |$

解题步骤 2.2

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

点击获取更多步骤...

解题步骤 2.2.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

解题步骤 2.2.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

解题步骤 2.2.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

解题步骤 2.2.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

解题步骤 2.2.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

解题步骤 2.2.6

Multiply element $a_{12}$ by its cofactor.

$2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

解题步骤 2.2.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

解题步骤 2.2.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

解题步骤 2.2.9

Add the terms together.

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

解题步骤 2.3

将 $0$ 乘以 $| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$ 。

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

解题步骤 2.4

计算 $| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$ 。

点击获取更多步骤...

解题步骤 2.4.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

$1 (1 \cdot 80000 - 40000 \cdot 1) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

解题步骤 2.4.2

化简行列式。

点击获取更多步骤...

解题步骤 2.4.2.1

化简每一项。

点击获取更多步骤...

解题步骤 2.4.2.1.1

将 $80000$ 乘以 $1$ 。

$1 (80000 - 40000 \cdot 1) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

解题步骤 2.4.2.1.2

将 $- 40000$ 乘以 $1$ 。

$1 (80000 - 40000) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

解题步骤 2.4.2.2

从 $80000$ 中减去 $40000$ 。

$1 \cdot 40000 + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

解题步骤 2.5

计算 $| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$ 。

点击获取更多步骤...

解题步骤 2.5.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

$1 \cdot 40000 + 2 (1 \cdot 80000 - 45000 \cdot 1) + 0$

解题步骤 2.5.2

化简行列式。

点击获取更多步骤...

解题步骤 2.5.2.1

化简每一项。

点击获取更多步骤...

解题步骤 2.5.2.1.1

将 $80000$ 乘以 $1$ 。

$1 \cdot 40000 + 2 (80000 - 45000 \cdot 1) + 0$

解题步骤 2.5.2.1.2

将 $- 45000$ 乘以 $1$ 。

$1 \cdot 40000 + 2 (80000 - 45000) + 0$

解题步骤 2.5.2.2

从 $80000$ 中减去 $45000$ 。

$1 \cdot 40000 + 2 \cdot 35000 + 0$

解题步骤 2.6

化简行列式。

点击获取更多步骤...

解题步骤 2.6.1

化简每一项。

点击获取更多步骤...

解题步骤 2.6.1.1

将 $40000$ 乘以 $1$ 。

$40000 + 2 \cdot 35000 + 0$

解题步骤 2.6.1.2

将 $2$ 乘以 $35000$ 。

$40000 + 70000 + 0$

解题步骤 2.6.2

将 $40000$ 和 $70000$ 相加。

$110000 + 0$

解题步骤 2.6.3

将 $110000$ 和 $0$ 相加。

$110000$

$D = 110000$

解题步骤 3

Since the determinant is not $0$ , the system can be solved using Cramer's Rule.

解题步骤 4

Find the value of $x$ by Cramer's Rule, which states that $x = \frac{D_{x}}{D}$ .

点击获取更多步骤...

解题步骤 4.1

Replace column $1$ of the coefficient matrix that corresponds to the $x$ -coefficients of the system with $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 0 & - 2 & 0 \\ 310 & 1 & 1 \\ 16000000 & 40000 & 80000 \end{array} |$

解题步骤 4.2

Find the determinant.

点击获取更多步骤...

解题步骤 4.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

点击获取更多步骤...

解题步骤 4.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

解题步骤 4.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

解题步骤 4.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

解题步骤 4.2.1.4

Multiply element $a_{11}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

解题步骤 4.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

解题步骤 4.2.1.6

Multiply element $a_{12}$ by its cofactor.

$2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

解题步骤 4.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

解题步骤 4.2.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

解题步骤 4.2.1.9

Add the terms together.

$0 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

解题步骤 4.2.2

将 $0$ 乘以 $| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$ 。

$0 + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

解题步骤 4.2.3

将 $0$ 乘以 $| \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$ 。

$0 + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0$

解题步骤 4.2.4

计算 $| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$ 。

点击获取更多步骤...

解题步骤 4.2.4.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

$0 + 2 (310 \cdot 80000 - 16000000 \cdot 1) + 0$

解题步骤 4.2.4.2

化简行列式。

点击获取更多步骤...

解题步骤 4.2.4.2.1

化简每一项。

点击获取更多步骤...

解题步骤 4.2.4.2.1.1

将 $310$ 乘以 $80000$ 。

$0 + 2 (24800000 - 16000000 \cdot 1) + 0$

解题步骤 4.2.4.2.1.2

将 $- 16000000$ 乘以 $1$ 。

$0 + 2 (24800000 - 16000000) + 0$

解题步骤 4.2.4.2.2

从 $24800000$ 中减去 $16000000$ 。

$0 + 2 \cdot 8800000 + 0$

解题步骤 4.2.5

化简行列式。

点击获取更多步骤...

解题步骤 4.2.5.1

将 $2$ 乘以 $8800000$ 。

$0 + 17600000 + 0$

解题步骤 4.2.5.2

将 $0$ 和 $17600000$ 相加。

$17600000 + 0$

解题步骤 4.2.5.3

将 $17600000$ 和 $0$ 相加。

$17600000$

$D_{x} = 17600000$

解题步骤 4.3

Use the formula to solve for $x$ .

$x = \frac{D_{x}}{D}$

解题步骤 4.4

Substitute $110000$ for $D$ and $17600000$ for $D_{x}$ in the formula.

$x = \frac{17600000}{110000}$

解题步骤 4.5

用 $17600000$ 除以 $110000$ 。

$x = 160$

解题步骤 5

Find the value of $y$ by Cramer's Rule, which states that $y = \frac{D_{y}}{D}$ .

点击获取更多步骤...

解题步骤 5.1

Replace column $2$ of the coefficient matrix that corresponds to the $y$ -coefficients of the system with $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 1 & 0 & 0 \\ 1 & 310 & 1 \\ 45000 & 16000000 & 80000 \end{array} |$

解题步骤 5.2

Find the determinant.

点击获取更多步骤...

解题步骤 5.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

点击获取更多步骤...

解题步骤 5.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

解题步骤 5.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

解题步骤 5.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

解题步骤 5.2.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

解题步骤 5.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

解题步骤 5.2.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

解题步骤 5.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

解题步骤 5.2.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

解题步骤 5.2.1.9

Add the terms together.

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

解题步骤 5.2.2

将 $0$ 乘以 $| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$ 。

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 + 0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

解题步骤 5.2.3

将 $0$ 乘以 $| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$ 。

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 + 0$

解题步骤 5.2.4

计算 $| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$ 。

点击获取更多步骤...

解题步骤 5.2.4.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

$1 (310 \cdot 80000 - 16000000 \cdot 1) + 0 + 0$

解题步骤 5.2.4.2

化简行列式。

点击获取更多步骤...

解题步骤 5.2.4.2.1

化简每一项。

点击获取更多步骤...

解题步骤 5.2.4.2.1.1

将 $310$ 乘以 $80000$ 。

$1 (24800000 - 16000000 \cdot 1) + 0 + 0$

解题步骤 5.2.4.2.1.2

将 $- 16000000$ 乘以 $1$ 。

$1 (24800000 - 16000000) + 0 + 0$

解题步骤 5.2.4.2.2

从 $24800000$ 中减去 $16000000$ 。

$1 \cdot 8800000 + 0 + 0$

解题步骤 5.2.5

化简行列式。

点击获取更多步骤...

解题步骤 5.2.5.1

将 $8800000$ 乘以 $1$ 。

$8800000 + 0 + 0$

解题步骤 5.2.5.2

将 $8800000$ 和 $0$ 相加。

$8800000 + 0$

解题步骤 5.2.5.3

将 $8800000$ 和 $0$ 相加。

$8800000$

$D_{y} = 8800000$

解题步骤 5.3

Use the formula to solve for $y$ .

$y = \frac{D_{y}}{D}$

解题步骤 5.4

Substitute $110000$ for $D$ and $8800000$ for $D_{y}$ in the formula.

$y = \frac{8800000}{110000}$

解题步骤 5.5

用 $8800000$ 除以 $110000$ 。

$y = 80$

解题步骤 6

Find the value of $z$ by Cramer's Rule, which states that $z = \frac{D_{z}}{D}$ .

点击获取更多步骤...

解题步骤 6.1

Replace column $3$ of the coefficient matrix that corresponds to the $z$ -coefficients of the system with $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 310 \\ 45000 & 40000 & 16000000 \end{array} |$

解题步骤 6.2

Find the determinant.

点击获取更多步骤...

解题步骤 6.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

点击获取更多步骤...

解题步骤 6.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

解题步骤 6.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

解题步骤 6.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$

解题步骤 6.2.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$

解题步骤 6.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

解题步骤 6.2.1.6

Multiply element $a_{12}$ by its cofactor.

$2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

解题步骤 6.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

解题步骤 6.2.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

解题步骤 6.2.1.9

Add the terms together.

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} | + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

解题步骤 6.2.2

将 $0$ 乘以 $| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$ 。

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} | + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

解题步骤 6.2.3

计算 $| \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$ 。

点击获取更多步骤...

解题步骤 6.2.3.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

$1 (1 \cdot 16000000 - 40000 \cdot 310) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

解题步骤 6.2.3.2

化简行列式。

点击获取更多步骤...

解题步骤 6.2.3.2.1

化简每一项。

点击获取更多步骤...

解题步骤 6.2.3.2.1.1

将 $16000000$ 乘以 $1$ 。

$1 (16000000 - 40000 \cdot 310) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

解题步骤 6.2.3.2.1.2

将 $- 40000$ 乘以 $310$ 。

$1 (16000000 - 12400000) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

解题步骤 6.2.3.2.2

从 $16000000$ 中减去 $12400000$ 。

$1 \cdot 3600000 + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

解题步骤 6.2.4

计算 $| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$ 。

点击获取更多步骤...

解题步骤 6.2.4.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

$1 \cdot 3600000 + 2 (1 \cdot 16000000 - 45000 \cdot 310) + 0$

解题步骤 6.2.4.2

化简行列式。

点击获取更多步骤...

解题步骤 6.2.4.2.1

化简每一项。

点击获取更多步骤...

解题步骤 6.2.4.2.1.1

将 $16000000$ 乘以 $1$ 。

$1 \cdot 3600000 + 2 (16000000 - 45000 \cdot 310) + 0$

解题步骤 6.2.4.2.1.2

将 $- 45000$ 乘以 $310$ 。

$1 \cdot 3600000 + 2 (16000000 - 13950000) + 0$

解题步骤 6.2.4.2.2

从 $16000000$ 中减去 $13950000$ 。

$1 \cdot 3600000 + 2 \cdot 2050000 + 0$

解题步骤 6.2.5

化简行列式。

点击获取更多步骤...

解题步骤 6.2.5.1

化简每一项。

点击获取更多步骤...

解题步骤 6.2.5.1.1

将 $3600000$ 乘以 $1$ 。

$3600000 + 2 \cdot 2050000 + 0$

解题步骤 6.2.5.1.2

将 $2$ 乘以 $2050000$ 。

$3600000 + 4100000 + 0$

解题步骤 6.2.5.2

将 $3600000$ 和 $4100000$ 相加。

$7700000 + 0$

解题步骤 6.2.5.3

将 $7700000$ 和 $0$ 相加。

$7700000$

$D_{z} = 7700000$

解题步骤 6.3

Use the formula to solve for $z$ .

$z = \frac{D_{z}}{D}$

解题步骤 6.4

Substitute $110000$ for $D$ and $7700000$ for $D_{z}$ in the formula.

$z = \frac{7700000}{110000}$

解题步骤 6.5

用 $7700000$ 除以 $110000$ 。

$z = 70$

解题步骤 7

列出方程组的解。

$x = 160$

$y = 80$

$z = 70$

有限数学 示例

有限数学示例