The problem asks us to solve the given system of equations: $4x - 2y = 2$ (1) $-32x + 16y = -16$ (2) If there is no unique solution, we must state a reason.

1. Problem Description

The problem asks us to solve the given system of equations:

$4x - 2y = 2$ (1)

$-32x + 16y = -16$ (2)

If there is no unique solution, we must state a reason.

2. Solution Steps

First, let's simplify the equations. We can divide the first equation by 2:

$2x - y = 1$ (1')

We can divide the second equation by -8:

$4x - 2y = 2$ (2')

Notice that equation (2') is exactly twice equation (1'). Let us check if equation (2) is a multiple of equation (1). If we multiply equation (1) by -8, we get:

$-8(4x - 2y) = -8(2)$

$-32x + 16y = -16$

Which is exactly equation (2). Therefore, the two equations are linearly dependent, meaning they represent the same line. This means there are infinitely many solutions.

We can express y in terms of x from equation (1'):

$2x - y = 1$

$y = 2x - 1$

Any ordered pair of the form $(x, 2x-1)$ is a solution to the system.

3. Final Answer

B. There are infinitely many solutions.