Questions: y=2-x/2 x=2y^2 A=

Solution Steps

To solve the given system of equations, we need to find the values of \(x\) and \(y\) that satisfy both equations. We can do this by substituting one equation into the other. Once we have the values of \(x\) and \(y\), we can use them to find the value of \(A\) if it is defined in terms of \(x\) and \(y\).

We start with the system of equations: \[ \begin{cases} y = 2 - \frac{x}{2} \\ x = 2y^2 \end{cases} \]

Substitute \( y = 2 - \frac{x}{2} \) into \( x = 2y^2 \): \[ x = 2 \left(2 - \frac{x}{2}\right)^2 \] Simplify and solve for \( x \): \[ x = 2 \left(4 - 2x + \frac{x^2}{4}\right) \] \[ x = 8 - 4x + \frac{x^2}{2} \] \[ \frac{x^2}{2} - 5x + 8 = 0 \] Multiply through by 2 to clear the fraction: \[ x^2 - 10x + 16 = 0 \] Solve the quadratic equation: \[ x = \frac{10 \pm \sqrt{100 - 64}}{2} \] \[ x = \frac{10 \pm 6}{2} \] \[ x = 8 \quad \text{or} \quad x = 2 \]

For \( x = 8 \): \[ y = 2 - \frac{8}{2} = 2 - 4 = -2 \] For \( x = 2 \): \[ y = 2 - \frac{2}{2} = 2 - 1 = 1 \]

Final Answer