The slope of parallel and perpendicular lines

Algebra-1

1. Fundamental Concepts

Definition: The slope of a line is a measure of its steepness, calculated as the change in y divided by the change in x ($$\frac{{\Delta y}}{{\Delta x}}$$).
Parallel Lines: Two lines are parallel if their slopes are equal ($$m_1 = m_2$$).
Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1 ($$m_1 \cdot m_2 = -1$$).

2. Key Concepts

Slope Formula: $$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$$

Parallel Lines Condition: If two lines are parallel, then $$m_1 = m_2$$.

Perpendicular Lines Condition: If two lines are perpendicular, then $$m_1 \cdot m_2 = -1$$.

3. Examples

Example 1 (Basic)

Problem: Determine if the lines with equations $$y = 2x + 3$$ and $$y = 2x - 4$$ are parallel.

Step-by-Step Solution:

Identify the slopes of both lines. For the first line, the slope is $$2$$.
For the second line, the slope is also $$2$$.
Since the slopes are equal ($$2 = 2$$), the lines are parallel.

Validation: Both lines have the same slope, confirming they are parallel.

Example 2 (Intermediate)

Problem: Determine if the lines with equations $$y = 3x + 5$$ and $$y = -\frac{1}{3}x + 2$$ are perpendicular.

Step-by-Step Solution:

Identify the slopes of both lines. For the first line, the slope is $$3$$.
For the second line, the slope is $$-\frac{1}{3}$$.
Calculate the product of the slopes: $$3 \cdot -\frac{1}{3} = -1$$.
Since the product of the slopes is $$-1$$, the lines are perpendicular.

Validation: The product of the slopes equals $$-1$$, confirming they are perpendicular.

4. Problem-Solving Techniques

Graphical Method: Plot the lines on a coordinate plane to visually determine if they are parallel or perpendicular.
Algebraic Verification: Use the slope formula to calculate the slopes and verify the conditions for parallelism and perpendicularity.
Substitution Check: Substitute points from one line into the equation of the other to check for consistency.