Determine if the lines $$y = -2x + 3$$ and $$y = 2x - 1$$ are parallel, perpendicular, or neither.

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To determine the relationship between the lines given by the equations $$y = -2x + 3$$ and $$y = 2x - 1$$, we need to analyze their slopes.

The slope-intercept form of a line is given by the equation $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.

1. For the first line $$y = -2x + 3$$:
   - The slope $$m_1$$ is $$-2$$.

2. For the second line $$y = 2x - 1$$:
   - The slope $$m_2$$ is $$2$$.

Next, we check the relationship between the slopes:

- Two lines are **parallel** if their slopes are equal, i.e., $$m_1 = m_2$$.
- Two lines are **perpendicular** if the product of their slopes is $$-1$$, i.e., $$m_1 \cdot m_2 = -1$$.

Now, we calculate the product of the slopes:

$$
m_1 \cdot m_2 = (-2) \cdot (2) = -4
$$

Since $$-4$$ is not equal to $$-1$$, the lines are not perpendicular. Additionally, since the slopes $$-2$$ and $$2$$ are not equal, the lines are not parallel either.

Thus, the conclusion is that the lines $$y = -2x + 3$$ and $$y = 2x - 1$$ are **neither parallel nor perpendicular**.
The lines $$y = -2x + 3$$ and $$y = 2x - 1$$ are neither parallel nor perpendicular.

Determine if the lines and are parallel, perpendicular, or neither.

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