91Ó°ÊÓ

A critical component of understanding parallel lines is the concept of the slope. The slope of a line, often represented by the letter \( m \), tells us how steep the line is. The slope is calculated as the change in the \( y \)-coordinates divided by the change in the \( x \)-coordinates between two points on a line. Simplified, the slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In the problem at hand, the slopes of both lines are equal to 1 (from the equations \( y = x + 2 \) and \( y = x - 4 \), since the line equations are in the form \( y = mx + b \)), which confirms they are parallel. When two lines have the same slope, it means they rise and run at the same rate, maintaining a consistent distance without ever intersecting.

Parallel lines are lines in a plane that never meet; they are always the same distance apart, maintaining this consistency along their entire length. This important geometric feature makes calculating the distance between them a straightforward exercise.

For any lines to be parallel:

• They must have the same slope.
• Their equations differ only by the constant term.

Given lines such as \( y = x + 2 \) and \( y = x - 4 \), the difference lies in the constant (known as the y-intercept). The y-intercepts do not affect the slope but indicate where each line crosses the y-axis. These lines are graphed in a straight fashion such that they never intersect.

Finding the distance between parallel lines involves specific formulas tailored for this purpose. The distance \( d \) between two parallel lines given by \( y = mx + c_1 \) and \( y = mx + c_2 \) can be calculated using:

• \( d = \frac{|c_2 - c_1|}{\sqrt{1 + m^2}} \)

This formula is derived from the general distance concept, adapted to suit the parallel nature.

In our problem, substituting \( m = 1 \), \( c_1 = 2 \), and \( c_2 = -4 \) gives:

• Numerator: The absolute difference \(|-4 - 2| = 6\).
• Denominator: Simplifying \( \sqrt{1 + 1^2} = \sqrt{2} \).

Inserting these values back into the formula gives \( d = \frac{6}{\sqrt{2}} \), and simplifying through rationalization results in \( 3\sqrt{2} \). Hence, the distance between the given lines is \( 3\sqrt{2} \), reflecting the uniform spacing characteristic of parallel lines.