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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The ksar is surrounded by barley fields and serves as a focal point for surrounding hamlets. Although the population has dwindled since the mid-20th century, traditional building techniques persist, with ongoing efforts to preserve the site.
Ksar Moulay Ali Chan emerged as a fortified settlement reflecting sedentary life and defensive needs in the Oubrovie region. Its architecture—with narrow, blending communal spaces en route to a central well—epitomizes the adaptation of desert-dwelling communities to environmental and social challenges. Built by the Murabit clan from the 13th century until the French colonial era, it embodies over 700 years of cultural continuity.
The mtaras inscription, which includes the ksar and over twenty similar ksour, recognizes communal kinship structures and mud architecture as vital expressions of North African identity. By incorporating fortified dwellings, oases, and trade routes, these sites illustrate sustainable land use in arid zones.

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📰 \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \frac{3}{2} 📰 Equality holds when \( x = y = z \). Since \( x + y + z = 6 \), set \( x = y = z = 2 \). 📰 \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}

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:

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