Graph parallel and perpendicular lines

Algebra-1

1. Fundamental Concepts

Definition: Parallel lines are lines in a plane that do not intersect, and their slopes are equal.
Definition: Perpendicular lines are lines that intersect at a right angle (90 degrees), and the product of their slopes is \(-1\).
Slope Formula: The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

2. Key Concepts

Slope of Parallel Lines: If two lines are parallel, then \(m_1 = m_2\).

Slope of Perpendicular Lines: If two lines are perpendicular, then \(m_1 \cdot m_2 = -1\).

Graphing Techniques: Use the slope-intercept form \(y = mx + b\) to graph lines.

3. Examples

Example 1 (Basic)

Problem: Graph the lines \(y = 2x + 3\) and \(y = 2x - 4\).

Step-by-Step Solution:

Identify the slope and y-intercept for each line. Both lines have a slope of \(2\).
Plot the y-intercepts: \((0, 3)\) and \((0, -4)\).
Use the slope to find another point on each line. For example, from \((0, 3)\), go up 2 units and right 1 unit to get \((1, 5)\).
Draw the lines through these points.

Validation: Check if the lines do not intersect.

Example 2 (Intermediate)

Problem: Graph the lines \(y = 3x + 2\) and \(y = -\frac{1}{3}x - 1\).

Step-by-Step Solution:

Identify the slopes: \(3\) and \(-\frac{1}{3}\). Since \(3 \cdot -\frac{1}{3} = -1\), the lines are perpendicular.
Plot the y-intercepts: \((0, 2)\) and \((0, -1)\).
Use the slopes to find another point on each line. For example, from \((0, 2)\), go up 3 units and right 1 unit to get \((1, 5)\).
Draw the lines through these points.

Validation: Check if the lines intersect at a right angle.

4. Problem-Solving Techniques

Visual Strategy: Use graph paper to accurately plot points and draw lines.
Error-Proofing: Double-check the slope calculations and ensure the lines are correctly plotted.
Concept Reinforcement: Practice identifying parallel and perpendicular lines from their equations.