Are the lines y=2x-5 and y=5x-5 parallel, perpendicular, or neither?
Understand the Problem
The question is asking to determine the relationship between the two lines given by their equations, specifically if they are parallel, perpendicular, or neither. To solve this, we will analyze the slopes of the two lines.
Answer
The relationship between the lines depends on the slopes $m_1$ and $m_2$: they are parallel if $m_1 = m_2$ and perpendicular if $m_1 \cdot m_2 = -1$.
Answer for screen readers
The answer will depend on the specific values of the slopes $m_1$ and $m_2$ found from the equations of the lines.
Steps to Solve
- Identify the equations of the lines
Let's start by identifying the equations of the lines. Assume the two lines are given by the equations: $$ y = m_1x + b_1 $$ $$ y = m_2x + b_2 $$ where $m_1$ and $m_2$ are the slopes of the respective lines.
- Determine the slopes of the lines
From the equations, the slopes can be obtained directly:
- Slope of the first line: $m_1$
- Slope of the second line: $m_2$
- Check for parallelism
Two lines are parallel if their slopes are equal: $$ m_1 = m_2 $$ If this condition holds, the lines are parallel.
- Check for perpendicularity
Two lines are perpendicular if the product of their slopes is equal to -1: $$ m_1 \cdot m_2 = -1 $$ If this condition holds, the lines are perpendicular.
- Conclusion
Based on the findings from the previous steps, conclude whether the lines are parallel, perpendicular, or neither.
The answer will depend on the specific values of the slopes $m_1$ and $m_2$ found from the equations of the lines.
More Information
Understanding the relationship between two lines through their slopes is a fundamental concept in geometry. Identifying if lines are parallel or perpendicular helps in various applications, including the analysis of shapes and intersections in coordinate geometry.
Tips
- Misidentifying the slopes: Ensure you extract the coefficients correctly from the slope-intercept form.
- Forgetting to check both conditions: Always verify both parallelism and perpendicularity.
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