x^2 + y^2 + z^2 - 2z + 1 - (x^2 - 2x + 1 + y^2 + z^2) = 0 - Crosslake
SEO-Optimized Article: Simplify and Solve the 3D Equation — x² + y² + z² − 2z + 1 − (x² − 2x + 1 + y² + z²) = 0
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.
Image Gallery
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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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:
\[
x - z = 0 \quad \ ext{or} \quad x = z
\]
