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

Understanding the Context

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.

Solved 1. y=x^2 - z2 2. x^2 - y^2 + z^2 = 1 3. y = 2x^2 + | Chegg.com

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

Key Insights

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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Solution: Let $\mathbf{v} = \begin{pmatrix} a \\ b \end{pmatrix}$ be any vector orthogonal to $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$, i.e.,
\mathbf{v} \cdot \begin{pmatrix} 3 \\ -4 \end{pmatrix} = 3a - 4
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Final Thoughts

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:

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