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The slope-intercept form of a linear equation is one of the most common ways to express equations of lines. It is presented as:
\( y = mx + b \)
Here:

• \( y \) represents the dependent variable or the vertical value on the graph.
• \( x \) is the independent variable or the horizontal value.
• \( m \) stands for the slope of the line.
• \( b \) is the y-intercept.

This form is especially useful because it immediately shows the slope and the y-intercept of the line, making it easier to graph and analyze. For example, rewriting the equation \( y - \frac{3}{2}x - 1 = 0 \) in slope-intercept form, we get \( y = \frac{3}{2}x + 1 \). This reveals that the slope is \( \frac{3}{2} \) and the y-intercept is 1.

The slope of a line measures its steepness and direction. It is represented by the letter \( m \) in the equation \( y = mx + b \). Mathematically, the slope can be defined as:
\( m = \frac{\text{rise}}{\text{run}} \)
This means:

• 'Rise' - The vertical change between two points.
• 'Run' - The horizontal change between the same two points.

In the example exercise, the equation is \( y = \frac{3}{2}x + 1 \). Here, the slope \( \frac{3}{2} \) indicates that for every 2 units you move horizontally to the right, you move 3 units up vertically. Positive slopes rise to the right, while negative slopes fall to the right.

The y-intercept of a line is where the line crosses the y-axis. It is represented by the letter \( b \) in the equation \( y = mx + b \). In other words, it is the value of \( y \) when \( x = 0 \).
For the equation \( y = \frac{3}{2}x + 1 \):

• Set \( x \) to 0
• The resulting equation is \( y = 1 \)

Therefore, the y-intercept is 1. This point gives a starting value for graphing, and you can begin plotting the line from this specific point on the y-axis.

Graphing lines using the slope-intercept form \( y = mx + b \) is straightforward and follows a pattern. Follow these steps:
1. Identify and plot the y-intercept \( b \). For example, (0, 1).
2. From the y-intercept, use the slope \( m \) to determine the next points. Given \( m = \frac{3}{2} \), move 3 units up and 2 units right from the y-intercept. Plot this second point.
3. Draw a straight line through these two points extended in both directions to complete the graph.
Using our example, starting at the y-intercept (0, 1), moving up 3 units to (0, 4), and 2 units to the right, you finish at the point (2, 4). Connect these points, and you have successfully graphed the line \( y = \frac{3}{2}x + 1 \).