x^2 + y^2 - 2y + 1 + z^2 - (x^2 - 2x + 1 + y^2 + z^2) = 0 - Aurero
Title: Simplifying and Interpreting the 3D Equation: A Comprehensive Guide to x² + y² − 2y + 1 + z² − (x² − 2x + 1 + y² + z²) = 0
Title: Simplifying and Interpreting the 3D Equation: A Comprehensive Guide to x² + y² − 2y + 1 + z² − (x² − 2x + 1 + y² + z²) = 0
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.
Understanding the Context
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
]
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Key Insights
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.
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:
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[
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
]
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).
