x^2 + y^2 - 2y + 1 + z^2 - (x^2 - 2x + 1 + y^2 + z^2) = 0 - Treasure Valley Movers

Title: Simplifying and Interpreting the 3D Equation: A Comprehensive Guide to x² + y² − 2y + 1 + z² − (x² − 2x + 1 + y² + z²) = 0

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Meta Description:
Explore the simplification and geometric meaning of the 3D equation x² + y² − 2y + 1 + z² − (x² − 2x + 1 + y² + z²) = 0. Discover how this equation describes a point in space and how to rewrite it in standard form.

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Introduction

Mathematical equations often encode rich geometric information, especially in three dimensions. Today, we analyze and simplify a key equation:

[
x^2 + y^2 - 2y + 1 + z^2 - (x^2 - 2x + 1 + y^2 + z^2) = 0
]

Under the hood, this equation represents a point in space—specifically, it reduces to a single coordinate condition, revealing a specific location in 3D geometry. Let’s break this down step by step.

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Step 1: Expand and Simplify the Expression

Start by expanding both sides of the equation. Note that the expression includes a parenthetical term:
[
-(x^2 - 2x + 1 + y^2 + z^2)
]

Distribute the negative sign:

[
x^2 + y^2 - 2y + 1 + z^2 - x^2 + 2x - 1 - y^2 - z^2 = 0
]

Now combine like terms:

• (x^2 - x^2 = 0)
  - (y^2 - y^2 = 0)
  - (z^2 - z^2 = 0)
  - (-2y) remains
  - (+1 - 1 = 0)
  - (+2x) remains

After cancellation, the entire left-hand side reduces to:

[
-2y + 2x = 0
]

So:

[
2x - 2y = 0 \quad \Rightarrow \quad x = y
]

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Step 2: Interpretation in 3D Space

At first glance, this appears degenerate—a 2D plane (x = y) extended along (z). However, note that the variables (z) and higher-degree terms canceled out completely, leaving only the condition (x = y), independent of (z) and (y).