Are the lines y = x - 4 + 1 and 4y = x + 3 parallel, perpendicular, or neither?
Understand the Problem
The question is asking whether the two given lines are parallel, perpendicular, or neither by analyzing their slopes. To determine this, we need to put both equations into slope-intercept form (y = mx + b) to find and compare their slopes.
Answer
The lines are perpendicular.
Answer for screen readers
The two lines are perpendicular.
Steps to Solve
- Identify the equations of the lines
Let's say we have two line equations:
$$ \text{Line 1: } y = 2x + 3 $$
$$ \text{Line 2: } y = -\frac{1}{2}x + 4 $$
- Extract the slopes
The slope-intercept form of a line is given as $y = mx + b$, where $m$ is the slope.
- For Line 1, the slope $m_1 = 2$.
- For Line 2, the slope $m_2 = -\frac{1}{2}$.
- Determine if the lines are parallel, perpendicular, or neither
- Parallel lines have the same slope: $m_1 = m_2$.
- Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$.
Now, we'll check these properties:
- Check for parallel: Is $2 = -\frac{1}{2}$? No, so they are not parallel.
- Check for perpendicular:
Calculate the product of the slopes:
$$ m_1 \cdot m_2 = 2 \cdot \left(-\frac{1}{2}\right) = -1 $$
Since the product equals -1, the lines are perpendicular.
The two lines are perpendicular.
More Information
Perpendicular lines indicate a right angle where they intersect, which can be helpful in various applications, such as geometry and physics.
Tips
- Confusing parallel and perpendicular slopes: Remember, parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.
- Not converting to slope-intercept form correctly: Ensure you maintain proper arithmetic and isolation of $y$ in the equations.
