If `w != 0 and w = 2x = sqrt(2)y`, what is the value `w - x` in terms of y?

Options

A. `2y`
B. `(1)/(sqrt2)y`
C. `sqrt(2y)`
D. `4/(sqrt2) y`

Answer

To solve the equation \( w \neq 0 \) and \( w = 2x = \sqrt{2}y \), we need to find the value of \( w - x \) in terms of \( y \).

### Step-by-step solution:

1. **Set up the equations**:
We have two equations from the problem statement:
\[ w = 2x \]
\[ w = \sqrt{2}y \]

2. **Equate the two expressions for \( w \)**:
Since both expressions are equal to \( w \), we can set them equal to each other:
\[ 2x = \sqrt{2}y \]

3. **Solve for \( x \)**:
To find \( x \), we can rearrange the equation:
\[ x = \frac{\sqrt{2}y}{2} \]

4. **Substitute \( x \) back into the equation for \( w \)**:
Now that we have \( x \), we can find \( w \):
\[ w = 2x = 2 \left(\frac{\sqrt{2}y}{2}\right) = \sqrt{2}y \]

5. **Calculate \( w - x \)**:
Now we can find \( w - x \):
\[ w - x = \sqrt{2}y - \frac{\sqrt{2}y}{2} \]

6. **Simplify \( w - x \)**:
To simplify this expression, we can factor out \( \sqrt{2}y \):
\[ w - x = \sqrt{2}y \left(1 - \frac{1}{2}\right) = \sqrt{2}y \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}y \]

### Final Answer:
Thus, the value of \( w - x \) in terms of \( y \) is:
\[ \frac{\sqrt{2}}{2}y \]