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

February 17, 2026

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

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.

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:

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

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:

\[2x - 2z = 0\]

Step 3: Final Simplification

Divide both sides by 2:

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

What Does This Equation Represent?

The simplified equation \( x = z \) defines a plane in 3D space — specifically, a vertical plane that cuts through the origin, running diagonally between the \( x \)- and \( z \)-axes, where every point has equal \( x \) and \( z \) coordinates.

This surface has no curvature; it’s a flat, two-dimensional subspace within the 3D coordinate system, extending infinitely in the \( y \)-direction.

Practical Implications & Applications

Coordinate Geometry: The equation models a plane perpendicular to the vector \( \langle 1, 0, -1 \rangle \).- Physics & Engineering: Used in transformation geometry, symmetry analysis, and equilibrium conditions.- Computer Graphics: Helps define transformation planes in 3D modeling and rendering.

Visualization Tip

Imagine slicing through a 3D grid: every line where \( x = z \) forms a straight diagonal plane slicing through the cube, intersecting both axes at equal values.

Summary

The original equation simplifies elegantly to:

\[x = z\]

This plane is a fundamental geometric entity, simple yet powerful—ideal for understanding symmetry and relationships in three-dimensional space.

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Optimize your understanding further by exploring how this plane relates to norm buildup, distance formulas, or coordinate transformations in future posts!

Meta Description:Simplify the 3D equation \(x^2 + y^2 + z^2 - 2z + 1 - (x^2 - 2x + 1 + y^2 + z^2) = 0\) step-by-step to reveal it represents the plane \(x = z\). Learn its geometric meaning and significance in coordinate geometry.

Keywords: x = z plane, 3D geometry, algebra simplification, coordinate plane, math solved, geometry explained

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