91Ó°ÊÓ

The slope of a line is a measure of its steepness and direction. When examining a line on a coordinate plane, the slope represents how much the line rises or falls vertically for each unit it moves horizontally. It's vital in determining not just the angle of a line, but also how two lines interact—whether they are parallel, intersect, or perpendicular.

The formula for the slope, represented as 'm', is given by the difference in y-coordinates divided by the difference in x-coordinates of two distinct points on the line:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
From this, you can quickly see that:

• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates the line falls.
• A slope of zero results in a horizontal line.
• An undefined slope corresponds to a vertical line.

In our current exercise, the slope calculated for the points (0, 2) and (2, 3) is \(\frac{1}{2}\), meaning for every two units the line moves to the right, it rises by one unit.

The point-slope form of a line's equation is a useful format when you know the slope of a line and one specific point through which the line passes. It's particularly handy for quickly forming the equation of a line, especially in situations where these two data points are the most direct information available.

The point-slope form is structured as follows:
\[y - y_1 = m(x - x_1)\]
Where:

• \(m\) is the slope of the line.
• \((x_1, y_1)\) is a point on the line.

This form gives you a direct way to substitute known values of the slope and a specific point to find the equation of the line. In our exercise, given the slope \(\frac{1}{2}\) and the point (0, 2), substituting into the point-slope form yields:
\[y - 2 = \frac{1}{2}(x - 0)\]
Ultimately simplifying to \[y - 2 = \frac{1}{2}x\]. This expression illustrates how quickly and efficiently a point and a slope can construct the framework of a line's equation.

The slope-intercept form is perhaps the most familiar form of a linear equation. It provides a direct way to identify both the slope of the line and the y-intercept at a glance, making it invaluable for graphing and quick analysis.

This form of a linear equation is structured as:

• \(y = mx + b\)

Where:

• \(m\) denotes the slope.
• \(b\) represents the y-intercept, which is the value where the line crosses the y-axis.

In the exercise, the equation was transformed from the point-slope form to the slope-intercept form by solving for y and simplifying the terms. After starting with:
\[y - 2 = \frac{1}{2}x\]
Adding 2 to both sides provides:
\[y = \frac{1}{2}x + 2\]
This results in a clear depiction of the line's slope \(\frac{1}{2}\) and its y-intercept at (0, 2). This form makes it easy to predict and plot the line's behavior and understand its interaction with other lines on a graph.