If $4x + y = 1 \text{ and } 4x - y$

. then the values of x and y respectively are-

Detailed Explanation

Consider the given system of equations:

$4x + y = 1 \quad \text{(1)}$$ $$4x - y \quad \text{(2)}$

We need to find the values of \( x \) and \( y \) that satisfy both equations.

First, let us derive the value of \( y \) in terms of \( x \) from Equation (1):

$ y = 1 - 4x \quad \text{(3)}$

Next, we substitute Equation (3) into Equation (2) to eliminate \( y \) and solve for \( x \):

Substitute \( y = 1 - 4x \) into Equation (2):

$4x - (1 - 4x) = 0$

Simplify the equation:

$4x - 1 + 4x = 0$$ $$8x - 1 = 0$

Solving for \( x \), we get:

$8x = 1$$ $$x = \frac{1}{2}$

Now that we have \( x \), substitute \( x = \frac{1}{2} \) back into Equation (3) to find \( y \):

$ y = 1 - 4\left(\frac{1}{2}\right) $$ $$ y = 1 - 2 $$ $$ y = -1 $

Thus, the values of \( x \) and \( y \) that satisfy both equations are $\( x = \frac{1}{2} \) $and$ \( y = -\frac{1}{2} \).$

Therefore, the correct answer is:

$\left( \frac{1}{2}, -\frac{1}{2} \right)$