The **aim of this question** is to learn about the **three-dimensional space** $ R^3 $ and **its subsets.**

The **three-dimensional space** can be represented with the help of **3-coordinates** in the cartesian system. Usually, these coordinates are **x, y, and z-coordinates**. The **subsets** of this three-dimensional space can be described with the help of **constraint equations** that restrict the **domain or range** of the space.

The **subset region can have three possibilities**. If all **three coordinates** are constrained and there is a definite unique solution to all of them, then the subset region represents **a point**. If **two of them are constrained** and the third one is open, then the subset region represents **a plane**. And if all of the axes have no unique solution under the given constraints, then the **subset region is also a three-dimensional space.**

The constraints that we use to find these subsets may be **equations or inequalities**. In the **case of inequalities**, we first find the constraint using the **borderline equation**, and then we apply the **inequality** condition to find the **region of interest.**

## Expert Answer

**Recall the given equation:**

\[ x \ = \ 10 \]

Now notice that $ R^3 $ is **three-dimensional space** and to describe a region in a three-dimensional space, **we need to put constraints** on all of the three cartesian coordinates. If we **constraint only one** of the coordinates and the other **two are unconstrained** (which is the case here), then the **resulting region may be a plane.**

In our case, the region represents a **plain that spans the y and z coordinates** from negative infinity to positive infinity. In short and simple words, the **equation represents a yz-plane that cuts the x-axis at x = 10 mark.**

## Numerical Result

The equation x = 10 represents a yz-plane in $ R^3 $ that cuts the x-axis at x = 10 mark.

## Example

**Describe the region bound by the following equations in $ R^3 $ space.**

\[ x^2 \ = \ 10 y \ … \ … \ … \ ( 1 ) \]

\[ y \ = \ 10 z \ … \ … \ … \ ( 2 ) \]

\[ z \ = \ 10 x \ … \ … \ … \ ( 3 ) \]

Substituting the **value of z** from equation (3) in equation (2):

\[ y \ = \ 10 (10x) \]

\[ \Rightarrow y \ = \ 100 x \ … \ … \ … \ ( 4 ) \]

Substituting the **value of y** from equation (4) in equation (1):

\[ x^2 \ = \ 10 ( 100x ) \]

\[ \Rightarrow x^2 \ = \ 1000 x \]

\[ \Rightarrow x \ = \ 1000 \]

**Substituting this value in equation (3) and equation (4):**

\[ y \ = \ 100 (1000) \]

\[ \Rightarrow y \ = \ \ 100000 \]

\[ z \ = \ 10 (1000) \]

\[ \Rightarrow z \ = \ 10000 \]

**Hence we have a point:**

( x, y, z ) = ( 1000, 100000, 10000 )

which **required region represented by the above equations** in $ R^3 $.