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
]

Solved D 1. x2 - y2 + z2 = 1 2. x2 + 4y2 + 9z2 = 1 3. – x2 | Chegg.com

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Solved D 1. x2 - y2 + z2 = 1 2. x2 + 4y2 + 9z2 = 1 3. – x2 | Chegg.com
Solved 1. x2−y2+z2=1 2. x2+4y2+9z2=1 3. 9x2+4y2+z2=1 4. | Chegg.com
Solved 1. y=x^2 - z2 2. x^2 - y^2 + z^2 = 1 3. y = 2x^2 + | Chegg.com
Solved 1/x2 + 1/y2 / 1/x2 - 1/y2 - 1/x2 - 1/y2 / 1/x2 + 1/y2 | Chegg.com
Solved z^2 = x^2 + y^2 _____ z^2 = -1 + x^2 + y^2 _____ | Chegg.com

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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Now, total odd primes: 1 → contributes 1 odd number
Odd non-primes: 0 → but wait: the two non-prime rolls: we need total odd count even → current odd count = 1 (odd prime) → so even number of odd terms → need an odd number of odd non-primes
Odd non-primes contribute one odd each, even non-primes none.

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Final Thoughts

[
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).