91Ó°ÊÓ

When we talk about parallel lines, we mean lines that never intersect each other. Think of two railroad tracks running side by side; they always stay the same distance apart and never meet.

In mathematics, parallel lines have an important property: they have the same slope. This means if you know the slope of one line, you automatically know the slope of any line parallel to it.

For example, if Line A has a slope of 3, then any line parallel to Line A will also have a slope of 3.

The slope of a line measures how steep the line is. It tells us how much the y-value (vertical change) changes for a given change in the x-value (horizontal change).

The slope is usually denoted as 'm' and can be calculated by using the formula: \ m = \frac{\Delta y}{\Delta x} \ where \( \Delta y \) represents the change in y, and \( \Delta x \) represents the change in x.

When you look at a line in the slope-intercept form, which is \ y = mx + b \, the coefficient of \( x \) (\( m \)) is the slope. In our given problem, the equation was \( y = 3x - 7 \), making the slope 3.

The y-intercept is the point where the line crosses the y-axis. This happens when the value of \( x \) is 0.

In the slope-intercept form of an equation (\( y = mx + b \)), the y-intercept is represented by the constant term \( b \).

For instance, if we look at our initial equation, \( y = 3x - 7 \), the y-intercept is -7, which means the line crosses the y-axis at (0, -7).

When we form a new line parallel to the given line but with a different y-intercept, we use the given y-intercept to form a new equation. If the y-intercept is 4, as given in our problem, the equation becomes \( y = 3x + 4 \).

A linear equation is an equation that makes a straight line when graphed. The most common form of a linear equation is the slope-intercept form, given by \ y = mx + b \, where \ m \ is the slope and \ b \ is the y-intercept.

Linear equations are easy to work with because they have constant rates of change, and their graphs are straight lines. There are other forms of linear equations, like the standard form \ Ax + By = C \, but the slope-intercept form is often the simplest and most straightforward to use.

In the original exercise, we found the equation of a line parallel to a given line and used the slope-intercept form to easily identify the slope and y-intercept: \( y = 3x + 4 \).