Impact of Rotation on Linear Equations: y 2x - 3

Besides understanding the fundamental properties of linear equations, it is equally important to explore how these equations behave under different transformations, such as rotation. In this article, we delve into an interesting scenario where we investigate the effect of rotating the origin by 90, 180, and 270 degrees on the linear equation ({y 2x - 3}). Our investigation aims to provide a deeper understanding of coordinate transformations and their impact on the slope and intercept of lines.

1. 90 Degrees Counterclockwise Rotation

When a point is rotated 90 degrees counterclockwise around the origin, its coordinates transform as follows:

(x, y rightarrow -y, x)

To understand the impact of this transformation on the equation y 2x - 3, we substitute (x -y) and (y x). Let's perform these substitutions step-by-step:

First, express the equation in terms of the new coordinates:
(-y 2(-y) - 3) Simplify the equation:
(-y -2y - 3) Re-arrange to solve for (y):
(-y 2y -3) (y -frac{1}{2}x - frac{3}{2})

Therefore, the new equation after a 90-degree counterclockwise rotation is:

y -frac{1}{2}x - frac{3}{2}

2. 180 Degrees Rotation

For a 180-degree rotation, the coordinates transform as:

(x, y rightarrow -x, -y)

Starting from the original line equation y 2x - 3, we substitute (x -x) and (y -y):
(-y 2(-x) - 3)

Simplify the equation to its final form:

(-y -2x - 3)

Solving for (y), we get:

y 2x - 3

So, the new equation after a 180-degree rotation remains the same as the original equation:

3. 270 Degrees Counterclockwise Rotation

For a 270-degree counterclockwise rotation, the transformation can be seen as a 90-degree clockwise rotation. The coordinates transform as:

(x, y rightarrow y, -x)

Using the same original equation y 2x - 3, we substitute (x -y) and (y x):
(-x 2(-y) - 3)

After simplification:

(-x -2y - 3)

Solving for (y), we get:

y -frac{1}{2}x frac{3}{2}

Thus, the new equation after a 270-degree counterclockwise rotation is:

y -frac{1}{2}x frac{3}{2}

Summary of New Equations

The effect of the rotations summarized are as follows:

90 degrees counterclockwise: y -frac{1}{2}x - frac{3}{2} 180 degrees: y 2x - 3 270 degrees counterclockwise: y -frac{1}{2}x frac{3}{2}

These transformations effectively change the slope and y-intercept of the line based on the rotation applied. Understanding these transformations is crucial in various fields, including geometry, computer science, and even machine learning applications where coordinate transformations are common.