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The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. It is structured as follows: \[ y = mx + b \]where:

• \( y \) is the dependent variable (usually representing the vertical axis).
• \( m \) is the slope of the line.
• \( x \) is the independent variable (usually representing the horizontal axis).
• \( b \) is the y-intercept, the point where the line intersects the y-axis.

This form is extremely useful because it provides a quick way to graph a line and understand its steepness and direction, as indicated by the slope \( m \). A positive slope means the line ascends from left to right, while a negative slope means it descends. By isolating \( y \) (as we did in step 4 of the solution to the original exercise), this form simplifies the process of understanding and drawing the line's path on a coordinate plane.

The point-slope form is another essential way of expressing linear equations. It is particularly handy when you know a specific point on a line and the line's slope. The equation is structured as:\[ y - y_1 = m(x - x_1) \]where:

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

This format allows you to plug in values directly and understand how the line behaves at specific coordinates. Once you have these values, substitution gives you the starting form of the line equation. Then, you can easily rearrange it into slope-intercept form for graphing or further analysis. In the given exercise, we used point-slope form to form the equation with the point \((-2, -1)\) and slope \( \frac{3}{2} \), making it straightforward to map the new line intersecting at that point.

Understanding the concept of a negative reciprocal is key when dealing with perpendicular lines. For any slope \( m \), the slope of a perpendicular line is found by taking the reciprocal of \( m \) and then changing its sign. Let's break this down:

• If the original slope \( m = \frac{a}{b} \), the perpendicular slope is \( -\frac{b}{a} \).
• If the original slope is negative, its perpendicular slope will be positive, and vice versa.

This principle ensures that the product of the slopes of two perpendicular lines will always equal \(-1\). For example, in the exercise, the original slope was \(-\frac{2}{3}\). Thus, the perpendicular slope became \(\frac{3}{2}\). Grasping the idea of negative reciprocity allows you to effortlessly determine how lines relate to and intersect each other at right angles on a coordinate plane.