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When talking about linear equations, understanding the **slope of a line** is crucial. The slope is a measure of how steep the line is and in which direction it leans. It tells us how much the line rises or falls as we move from left to right on the graph. The generic formula for the slope, often represented as \( m \), is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of any two points on the line. This formula helps us calculate the change in \( y \) with respect to the change in \( x \).

• A positive slope means the line rises as it moves to the right.
• A negative slope indicates the line falls as it moves to the right.
• A slope of zero means the line is perfectly horizontal.
• An undefined slope implies a vertical line.

So in the exercise, the slope calculated was \( \frac{1}{2} \), meaning for every 2 units the line moves horizontally, it rises 1 unit vertically.

The **point-slope form** of a linear equation is very handy when you have a point on the line and the slope available. It is generally used to write the equation of a line quickly when these details are known. The formula for point-slope form is:\[ y - y_1 = m(x - x_1) \]- \( m \) is the slope of the line.- \( (x_1, y_1) \) is a specific known point on the line.This form allows you to plug in values directly and obtain the line's equation without much fuss. In the given problem, you used the slope \( \frac{1}{2} \) and the point \((-2, 1)\) to create the equation \( y - 1 = \frac{1}{2}(x + 2) \). This is particularly helpful because it's a straightforward step before converting the equation into other forms like the slope-intercept form.

The **slope-intercept form** is one of the most popular ways to express a linear equation because it's easy to interpret. The general structure looks like this:\[ y = mx + b \]- \( m \) is the slope of the line.- \( b \) is the y-intercept, indicating the point where the line crosses the y-axis.Using the slope-intercept form is advantageous when graphing a line, as it gives an immediate sense of how the line behaves:

• It quickly reveals the y-intercept value (\( b \)).
• The slope \( m \) shows the direction and steepness.

In the last step of the solution, by rearranging and simplifying the point-slope equation, we transformed it into the slope-intercept form: \( y = \frac{1}{2}x + 2 \). Here, \( \frac{1}{2} \) is the slope, and \( 2 \) is the y-intercept. This form is especially useful for identifying how the line will look on a graph.