The problem asks to find the transformed equation of the line $y + 2x - 4 = 0$ after being rotated by $-90^\circ$ around the point $(0, 0)$.

Step-by-step solution:

1. Original Line Equation: The equation of the line is: $y + 2x - 4 = 0 \quad \text{or} \quad y = -2x + 4$

2. Rotation by $-90^\circ$: The rotation matrix for a $-90^\circ$ rotation is:

   x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$ This gives the transformations: $$x' = y \quad \text{and} \quad y' = -x$$

3. Substitute the transformations: Replace $x$ and $y$ in the original equation with the new coordinates $x' = y$ and $y' = -x$: $(-x') + 2(y') - 4 = 0$ Simplifying this: $-x' + 2y' = 4$ So, the transformed equation of the line after a $-90^\circ$ rotation is: $-x' + 2y' = 4$

Would you like further details or have any questions?

Related Questions:

1. How does the transformation matrix for $90^\circ$ rotation differ?
2. How do you generalize the rotation of lines for any angle?
3. What happens if the rotation occurs around a point other than the origin?
4. How do reflections differ from rotations in terms of transformation matrices?
5. How would the equation change if rotated by $180^\circ$?

Tip: Always check the signs in rotation transformations, especially when dealing with negative angles, as they reverse direction!