Are the lines y=-1/3x+2 and y=3x-5 parallel, perpendicular, or neither?
Understand the Problem
The question is asking us to determine the relationship between two lines represented by their equations, specifically whether they are parallel, perpendicular, or neither. To solve this, we can compare the slopes of the lines. Two lines are parallel if their slopes are equal, and they are perpendicular if the product of their slopes is -1.
Answer
Determine the relationship based on the slopes: parallel if $m_1 = m_2$, perpendicular if $m_1 \cdot m_2 = -1$, or neither if not.
Answer for screen readers
The answer will depend on the specific values of the slopes of the two lines.
- If $m_1 = m_2$, the lines are parallel.
- If $m_1 \cdot m_2 = -1$, the lines are perpendicular.
- If both conditions are not satisfied, the lines are neither.
Steps to Solve
- Identify the equations of the lines
First, write down the equations of the two lines you are comparing. For example, let's say we have the lines:
Line 1: $y = m_1x + b_1$
Line 2: $y = m_2x + b_2$
- Determine the slopes of the lines
From the equations, identify the slopes ($m_1$ and $m_2$). The slope is the coefficient of $x$ in the slope-intercept form of the equation (i.e., $y = mx + b$).
- Compare the slopes
Next, compare the two slopes:
- If $m_1 = m_2$, the lines are parallel.
- If $m_1 \cdot m_2 = -1$, the lines are perpendicular.
- If neither of the above conditions is met, the lines are neither parallel nor perpendicular.
- Provide the relationship based on the comparison
Based on your comparison of the slopes, conclude whether the lines are parallel, perpendicular, or neither.
The answer will depend on the specific values of the slopes of the two lines.
- If $m_1 = m_2$, the lines are parallel.
- If $m_1 \cdot m_2 = -1$, the lines are perpendicular.
- If both conditions are not satisfied, the lines are neither.
More Information
Understanding the relationship between lines is fundamental in geometry and algebra. This knowledge is often used in various fields including physics and engineering. Relationships between lines help determine how shapes intersect and how to solve geometric problems.
Tips
- Misidentifying the slope: Ensure you correctly identify which coefficients correspond to $m_1$ and $m_2$ from the equations.
- Forgetting to check both conditions: It’s essential to check if the slopes are equal (for parallel) or if their product is -1 (for perpendicular). Both conditions should be tested.
- Incorrectly interpreting the results: Be careful when stating the relationship. Double-check your calculations.
