有限数学 示例

使用一个矩阵和克莱姆法则来求解。 x-2y=0 , x+y+z=310 , 45000x+40000y+80000z=16000000
, ,
解题步骤 1
以矩阵形式表示方程组。
解题步骤 2
Find the determinant of the coefficient matrix .
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解题步骤 2.1
Write in determinant notation.
解题步骤 2.2
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
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解题步骤 2.2.1
Consider the corresponding sign chart.
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 is the determinant with row and column deleted.
解题步骤 2.2.4
Multiply element by its cofactor.
解题步骤 2.2.5
The minor for is the determinant with row and column deleted.
解题步骤 2.2.6
Multiply element by its cofactor.
解题步骤 2.2.7
The minor for is the determinant with row and column deleted.
解题步骤 2.2.8
Multiply element by its cofactor.
解题步骤 2.2.9
Add the terms together.
解题步骤 2.3
乘以
解题步骤 2.4
计算
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解题步骤 2.4.1
可以使用公式 矩阵的行列式。
解题步骤 2.4.2
化简行列式。
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解题步骤 2.4.2.1
化简每一项。
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解题步骤 2.4.2.1.1
乘以
解题步骤 2.4.2.1.2
乘以
解题步骤 2.4.2.2
中减去
解题步骤 2.5
计算
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解题步骤 2.5.1
可以使用公式 矩阵的行列式。
解题步骤 2.5.2
化简行列式。
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解题步骤 2.5.2.1
化简每一项。
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解题步骤 2.5.2.1.1
乘以
解题步骤 2.5.2.1.2
乘以
解题步骤 2.5.2.2
中减去
解题步骤 2.6
化简行列式。
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解题步骤 2.6.1
化简每一项。
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解题步骤 2.6.1.1
乘以
解题步骤 2.6.1.2
乘以
解题步骤 2.6.2
相加。
解题步骤 2.6.3
相加。
解题步骤 3
Since the determinant is not , the system can be solved using Cramer's Rule.
解题步骤 4
Find the value of by Cramer's Rule, which states that .
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解题步骤 4.1
Replace column of the coefficient matrix that corresponds to the -coefficients of the system with .
解题步骤 4.2
Find the determinant.
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解题步骤 4.2.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
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解题步骤 4.2.1.1
Consider the corresponding sign chart.
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 is the determinant with row and column deleted.
解题步骤 4.2.1.4
Multiply element by its cofactor.
解题步骤 4.2.1.5
The minor for is the determinant with row and column deleted.
解题步骤 4.2.1.6
Multiply element by its cofactor.
解题步骤 4.2.1.7
The minor for is the determinant with row and column deleted.
解题步骤 4.2.1.8
Multiply element by its cofactor.
解题步骤 4.2.1.9
Add the terms together.
解题步骤 4.2.2
乘以
解题步骤 4.2.3
乘以
解题步骤 4.2.4
计算
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解题步骤 4.2.4.1
可以使用公式 矩阵的行列式。
解题步骤 4.2.4.2
化简行列式。
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解题步骤 4.2.4.2.1
化简每一项。
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解题步骤 4.2.4.2.1.1
乘以
解题步骤 4.2.4.2.1.2
乘以
解题步骤 4.2.4.2.2
中减去
解题步骤 4.2.5
化简行列式。
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解题步骤 4.2.5.1
乘以
解题步骤 4.2.5.2
相加。
解题步骤 4.2.5.3
相加。
解题步骤 4.3
Use the formula to solve for .
解题步骤 4.4
Substitute for and for in the formula.
解题步骤 4.5
除以
解题步骤 5
Find the value of by Cramer's Rule, which states that .
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解题步骤 5.1
Replace column of the coefficient matrix that corresponds to the -coefficients of the system with .
解题步骤 5.2
Find the determinant.
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解题步骤 5.2.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
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解题步骤 5.2.1.1
Consider the corresponding sign chart.
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 is the determinant with row and column deleted.
解题步骤 5.2.1.4
Multiply element by its cofactor.
解题步骤 5.2.1.5
The minor for is the determinant with row and column deleted.
解题步骤 5.2.1.6
Multiply element by its cofactor.
解题步骤 5.2.1.7
The minor for is the determinant with row and column deleted.
解题步骤 5.2.1.8
Multiply element by its cofactor.
解题步骤 5.2.1.9
Add the terms together.
解题步骤 5.2.2
乘以
解题步骤 5.2.3
乘以
解题步骤 5.2.4
计算
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解题步骤 5.2.4.1
可以使用公式 矩阵的行列式。
解题步骤 5.2.4.2
化简行列式。
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解题步骤 5.2.4.2.1
化简每一项。
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解题步骤 5.2.4.2.1.1
乘以
解题步骤 5.2.4.2.1.2
乘以
解题步骤 5.2.4.2.2
中减去
解题步骤 5.2.5
化简行列式。
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解题步骤 5.2.5.1
乘以
解题步骤 5.2.5.2
相加。
解题步骤 5.2.5.3
相加。
解题步骤 5.3
Use the formula to solve for .
解题步骤 5.4
Substitute for and for in the formula.
解题步骤 5.5
除以
解题步骤 6
Find the value of by Cramer's Rule, which states that .
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解题步骤 6.1
Replace column of the coefficient matrix that corresponds to the -coefficients of the system with .
解题步骤 6.2
Find the determinant.
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解题步骤 6.2.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
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解题步骤 6.2.1.1
Consider the corresponding sign chart.
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 is the determinant with row and column deleted.
解题步骤 6.2.1.4
Multiply element by its cofactor.
解题步骤 6.2.1.5
The minor for is the determinant with row and column deleted.
解题步骤 6.2.1.6
Multiply element by its cofactor.
解题步骤 6.2.1.7
The minor for is the determinant with row and column deleted.
解题步骤 6.2.1.8
Multiply element by its cofactor.
解题步骤 6.2.1.9
Add the terms together.
解题步骤 6.2.2
乘以
解题步骤 6.2.3
计算
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解题步骤 6.2.3.1
可以使用公式 矩阵的行列式。
解题步骤 6.2.3.2
化简行列式。
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解题步骤 6.2.3.2.1
化简每一项。
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解题步骤 6.2.3.2.1.1
乘以
解题步骤 6.2.3.2.1.2
乘以
解题步骤 6.2.3.2.2
中减去
解题步骤 6.2.4
计算
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解题步骤 6.2.4.1
可以使用公式 矩阵的行列式。
解题步骤 6.2.4.2
化简行列式。
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解题步骤 6.2.4.2.1
化简每一项。
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解题步骤 6.2.4.2.1.1
乘以
解题步骤 6.2.4.2.1.2
乘以
解题步骤 6.2.4.2.2
中减去
解题步骤 6.2.5
化简行列式。
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解题步骤 6.2.5.1
化简每一项。
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解题步骤 6.2.5.1.1
乘以
解题步骤 6.2.5.1.2
乘以
解题步骤 6.2.5.2
相加。
解题步骤 6.2.5.3
相加。
解题步骤 6.3
Use the formula to solve for .
解题步骤 6.4
Substitute for and for in the formula.
解题步骤 6.5
除以
解题步骤 7
列出方程组的解。