Parallel Lines: Mathematical Relationships and Properties

1. Slope-Intercept Form: The equation of a line can be written as y = mx + b, where m is the slope and b is the y-intercept.

2. Point-Slope Form: The equation of a line through a point (x₁, y₁) with slope m is y – y₁ = m(x – x₁).

3. Parallel Lines and Slope: Two lines are parallel if and only if they have the same slope. If l: y = m₁x + b₁ and m: y = m₂x + b₂, then l ∥ m if m₁ = m₂.

4. Perpendicular Slope: If two lines are perpendicular, the slope of one line is the negative reciprocal of the other. If one line has slope m, the other has slope -1/m (if m ≠ 0).

5. Parallel Lines in Coordinate Geometry: In the coordinate plane, two lines are parallel if their slopes are equal. The slope of a line through points (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁) / (x₂ – x₁).

Example of Two Parallel Lines:

1. Equation of the first line: y = 3x + 2

2. Equation of the second line: y = 3x – 4

Both lines have the same slope of 3, but different y-intercepts (2 and -4, respectively), making them parallel to each other.

Example of Two Parallel Lines with Fractional Slopes:

1. Equation of the first line: y = 2/3x + 1/2

2. Equation of the second line: y = 2/3x – 5/3

Both lines have the same slope of 2/3, but different y-intercepts, making them parallel to each other.

Example of Two Parallel Lines Given by Points:

Line 1: Points A(1, 2) and B(4, 5)

For Line 1: x₁ = 1, y₁ = 2, x₂ = 4, y₂ = 5

Slope of Line 1: m₁ = (y₂ – y₁) / (x₂ – x₁) = (5 – 2) / (4 – 1) = 3 / 3 = 1

Line 2: Points C(2, 3) and D(5, 6)

For Line 2: x₁ = 2, y₁ = 3, x₂ = 5, y₂ = 6

Slope of Line 2: m₂ = (y₂ – y₁) / (x₂ – x₁) = (6 – 3) / (5 – 2) = 3 / 3 = 1

Both lines have the same slope of 1, making them parallel to each other.