Are the lines y = 5/6x - 6 and x + 5y = 4 parallel, perpendicular, or neither?
Understand the Problem
The question is asking to determine the relationship between two lines defined by their equations, specifically whether they are parallel, perpendicular, or neither. This involves finding the slopes of both lines and comparing them.
Answer
The relationship between the lines is determined by their slopes: parallel if slopes are equal, perpendicular if the product is -1, otherwise neither.
Answer for screen readers
The relationship between the two lines depends on their slopes. If they are equal, they are parallel; if their product is -1, they are perpendicular; otherwise, they are neither.
Steps to Solve
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Identify the equations of the lines Given two line equations, for example, $y = mx + b$, we need to extract the slopes.
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Determine the slope of the first line For a line in the form $y = m_1x + b_1$, the slope $m_1$ is the coefficient of $x$.
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Determine the slope of the second line For a line in the form $y = m_2x + b_2$, the slope $m_2$ is also the coefficient of $x$.
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Compare the slopes to determine the relationship
- If $m_1 = m_2$, the lines are parallel.
- If $m_1 \cdot m_2 = -1$, the lines are perpendicular.
- If neither condition is satisfied, the lines are neither parallel nor perpendicular.
- Provide the conclusion Based on the comparison, state whether the lines are parallel, perpendicular, or neither.
The relationship between the two lines depends on their slopes. If they are equal, they are parallel; if their product is -1, they are perpendicular; otherwise, they are neither.
More Information
This problem revolves around understanding the slopes of linear equations. The concepts of parallel and perpendicular lines in geometry rely on slope comparisons, making it a fundamental topic in algebra and geometry.
Tips
- Confusing the slope of the equation. Remember that the slope is always associated with the coefficient of $x$.
- Failing to check both conditions for parallel and perpendicular lines. Always verify both the equality and the product of the slopes.
