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

Transcript text: Solve the following system of equations. \[ \begin{array}{l} 0.2 x+1.4 y=9 \\ 0.8 x-0.4 y=-6 \end{array} \] \[ \begin{array}{l} x= \\ y= \end{array} \]

Solution

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.

Step 1: Write the System of Equations

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

Step 2: Represent the System in Matrix Form

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

Step 3: Solve the Matrix Equation

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

Step 4: Extract the Solution

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