Questions: y=x+4z-5 4x+3y-2z=5 z=-2x+2

Solution Steps

To solve this system of linear equations, we can use matrix operations. We will represent the system in matrix form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the column matrix of constants. We can then use numpy's linear algebra solver to find the values of \(x\), \(y\), and \(z\).

The given system of equations is: \[ \begin{align_} y &= x + 4z - 5 \\ 4x + 3y - 2z &= 5 \\ z &= -2x + 2 \end{align_} \]

We can represent this system in matrix form \(AX = B\), where: \[ A = \begin{bmatrix} 1 & 0 & 4 \\ 4 & 3 & -2 \\ -2 & 0 & 1 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} -5 \\ 5 \\ 2 \end{bmatrix} \]

To find the values of \(x\), \(y\), and \(z\), we solve the matrix equation \(AX = B\). The solution to this equation is given by: \[ X = A^{-1}B \]

The solution to the matrix equation provides the values of the variables: \[ X = \begin{bmatrix} -1.4444 \\ 3.0000 \\ -0.8889 \end{bmatrix} \]

This corresponds to: \[ x = -1.4444, \quad y = 3.0000, \quad z = -0.8889 \]

Final Answer