Question

X+y+z=4 {x+y-z=2 2x-y+z=1 Решите систему уравнений

Answer 1

- X - Y - Z = - 4 
X + Y - Z = 2 
- 2Z = - 2 
Z = 1 
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X + Y = 4 - 1 
X + Y = 3 
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2X - Y = 1 - 1 
2X - Y = 0 
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X + Y = 3 
Y = 3 - X 
2X - ( 3 - X) = 0 
2X - 3 + X = 0 
3X = 3 
X = 1 
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1 + Y = 3 
Y = 3 - 1 
Y = 2 
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Ответ X = 1; Y = 2 ; Z = 1 
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Проверка
1) 1 + 2 + 1 = 4 ; 4 = 4 
2) 1 + 2 - 1 = 2 ; 2 = 2 
3) 2 - 2 + 1 = 1 ; 1 = 1 

Answer 2

Solving the System of Equations

To solve the system of equations x + y + z = 4, x + y - z = 2, and 2x - y + z = 1, we can use the method of substitution or elimination. Another approach is to use the Cramer's rule, which involves finding the determinants of the coefficient matrix and the matrices obtained by replacing one column with the constant terms. Let's proceed with Cramer's rule to solve this system of equations.

Cramer's Rule

Cramer's rule states that for a system of linear equations Ax = b, if the determinant of the coefficient matrix A is non-zero, then the system has a unique solution, and the solution can be expressed as x = D_x / D, where D_x is the determinant of the matrix obtained by replacing the x column of A with b, and D is the determinant of A.

Finding the Determinants

First, we need to find the determinant of the coefficient matrix. The coefficient matrix for the given system of equations is: ``` 1 1 1 1 1 -1 2 -1 1 ``` The determinant of this matrix can be found using the cofactor expansion method or any other suitable method.

Using Cramer's Rule

After finding the determinant of the coefficient matrix, we can proceed to find the determinants obtained by replacing the x, y, and z columns with the constants from the system of equations. Then, we can use Cramer's rule to find the values of x, y, and z.

Conclusion

By applying Cramer's rule, we can find the unique solution to the given system of equations x + y + z = 4, x + y - z = 2, and 2x - y + z = 1. This method provides a systematic approach to solving systems of linear equations and can be particularly useful when dealing with small systems like the one provided.