If two lines are parallel, their slopes are equal.
Understand the Problem
The question is asking about the relationship between the slopes of two parallel lines. Specifically, it seeks to clarify that if two lines are parallel, they have equal slopes.
Answer
The slopes of parallel lines are equal: $m_1 = m_2$.
Answer for screen readers
If two lines are parallel, their slopes are equal.
Steps to Solve
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Define the concept of parallel lines
Parallel lines are lines in a plane that never meet and are always the same distance apart. In terms of their slope, if two lines are parallel, they will have the same slope. -
Understand slope in linear equations
The slope of a line can be found from its equation in the form $y = mx + b$, where $m$ represents the slope and $b$ is the y-intercept. -
Establish the relationship between the slopes
If we have two linear equations:
Line 1: $y = m_1x + b_1$
Line 2: $y = m_2x + b_2$
For the lines to be parallel, it must hold that $m_1 = m_2$. -
Example for clarification
For instance, consider the lines $y = 2x + 3$ and $y = 2x - 5$.
Here, the slope $m_1 = 2$ and $m_2 = 2$.
Since $m_1 = m_2$, the lines are parallel.
If two lines are parallel, their slopes are equal.
More Information
Parallel lines will never intersect, and this characteristic is a foundation in geometry and analytic geometry. Understanding that slopes represent the steepness and direction of lines is crucial in graphing linear equations.
Tips
- Confusing parallel lines with lines that meet at an angle. Always remember that parallel lines do not intersect.
- Incorrectly calculating the slope. Ensure that slope is correctly determined using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ from two points on the line.
