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

SEO-Optimized Article: Simplify and Solve the 3D Equation — x² + y² + z² − 2z + 1 − (x² − 2x + 1 + y² + z²) = 0

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Solving the 3D Algebraic Mystery: A Step-by-Step Guide

Have you ever faced a seemingly complex equation involving multiple variables in 3D space and wondered how to simplify and interpret it? Today, we unravel the equation:

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

This expression appears in coordinate geometry, physics, and engineering, often representing surfaces in 3D space such as spheres. Let’s simplify, interpret, and visualize it.

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Step 1: Expand the Expression

First, expand the parentheses using the subtraction:

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

Distribute the negative sign:

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

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Step 2: Combine Like Terms

Group like terms:

• \( x^2 - x^2 = 0 \)
  - \( y^2 - y^2 = 0 \)
  - \( z^2 - z^2 = 0 \)
  - Constant terms: \( 1 - 1 = 0 \)
  - Remaining terms: \( -2z + 2x \)

So the equation simplifies to:

\[
2x - 2z = 0
\]

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Step 3: Final Simplification

Divide both sides by 2:

\[
x - z = 0 \quad \ ext{or} \quad x = z
\]