Questions: Solve the system of linear equations for x and y. (cos θ) x + (sin θ) y = 1 (-sin θ) x + (cos θ) y = 0 (x, y) = (sqrt(2)/2, sqrt(2)/2)

Solution Steps

We start with the given system of linear equations: \[ \begin{array}{r} (\cos \theta) x + (\sin \theta) y = 1 \\ (-\sin \theta) x + (\cos \theta) y = 0 \end{array} \] For \(\theta = \frac{\pi}{4}\), we substitute the values of \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).

The coefficient matrix \(A\) and the constants matrix \(B\) are defined as follows: \[ A = \begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \]

To find the values of \(x\) and \(y\), we solve the matrix equation \(AX = B\). The solution yields: \[ X = \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} \end{bmatrix} \]

Thus, we find that \(x = \frac{\sqrt{2}}{2}\) and \(y = \frac{\sqrt{2}}{2}\).

Final Answer