Questions: Solve the following system of equations. 0.2 x + 1.4 y = 9 0.8 x - 0.4 y = -6 x = y =

Solution Steps

To solve the system of linear equations with decimal coefficients, we can use matrix methods such as Gaussian elimination or the inverse matrix method. Here, we will use NumPy's linalg.solve function, which is designed to solve systems of linear equations efficiently.

We start with the given system of linear equations: \[ \begin{array}{l} 0.2x + 1.4y = 9 \\ 0.8x - 0.4y = -6 \end{array} \]

We can represent the system of equations in matrix form as \(A \mathbf{x} = B\), where: \[ A = \begin{bmatrix} 0.2 & 1.4 \\ 0.8 & -0.4 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 9 \\ -6 \end{bmatrix} \]

To find \(\mathbf{x}\), we solve the matrix equation \(A \mathbf{x} = B\). The solution is: \[ \mathbf{x} = A^{-1} B \]

The solution to the system of equations is: \[ \mathbf{x} = \begin{bmatrix} -4 \\ 7 \end{bmatrix} \] Thus, we have: \[ x = -4, \quad y = 7 \]

Final Answer