Understanding the Equation: x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0

The equation
x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0
might look complex at first glance, but it represents a meaningful geometric relationship in three-dimensional space. This article breaks down the equation, simplifies it, explores its geometric interpretation, and explains its relevance in applied mathematics and problem-solving.

Understanding the Context

Simplifying the Equation

Start with:
x² + (y - 1)² + z² – [(x - 1)² + y² + z²] = 0

First, expand each term carefully:

  1. Expand (y - 1)²:
    (y - 1)² = y² - 2y + 1
    So, x² + y² - 2y + 1 + z²
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 z^2 = x^2 + y^2 _____ z^2 = -1 + x^2 + y^2 _____ | Chegg.com
Solved x2+y2=1 x2+y2+z2=1 z=x+y z=x2−y2 z=x2 z2=x2+y2 | Chegg.com
Solved 1. x2−y2+z2=1 2. x2+4y2+9z2=1 3. 9x2+4y2+z2=1 4. | Chegg.com
Solved A D 1. −x2+y2−z2=1 2. x2−y2+z2=1 3. 9x2+4y2+z2=1 4. | Chegg.com

Key Insights

  1. Expand (x - 1)²:
    (x - 1)² = x² - 2x + 1
    So, (x² - 2x + 1) + y² + z²

Now substitute both into the original expression:

\[
[x² + y² - 2y + 1 + z²] - [x² - 2x + 1 + y² + z²] = 0
\]

Distribute the minus sign:

\[
x² + y² - 2y + 1 + z² - x² + 2x - 1 - y² - z² = 0
\]

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Final Thoughts

Now combine like terms:

  • \(x² - x² = 0\)
    - \(y² - y² = 0\)
    - \(z² - z² = 0\)
    - \(-2y + 2x + 1 - 1 = 2x - 2y\)

So the simplified equation is:
2x - 2y = 0
which further reduces to:
x – y = 0
or equivalently:
x = y

Geometric Interpretation

The simplified equation x = y describes a plane in three-dimensional space where the x-coordinate equals the y-coordinate. This is a vertical plane that slices through all values of z, passing diagonally across the xy-plane along the line where x = y.

Visualize shifting the classic planes. Instead of aligning with coordinate axes, this plane cuts diagonally from the origin along where x equals y, forming a “square-like” diagonal across quadrants where x and y are equal in magnitude and sign.

This plane is fundamental in symmetry considerations and acts as a decision boundary in optimization, machine learning, and physics problems involving diagonal variation or equilibrium.

Applications and Relevance