Linear Equations, Lines, Planes and Hyper-Planes

1. Any given equation in the following formats
   
   \(Ax + B=0\)   ...(1)
   
   \(Ax + By + C=0\)   ...(2)
   
   \(Ax + By + Cz +D=0\)   ...(3)
   
   \(Ax + By + Cz + + Dw + E=0\)   ...(4)
   
   is a Linear Equation. As given above, equation (1) is a Linear Equation of 1 Variable \(x\). Equation (2) is a Linear Equation of 2 Variables \(x\) and \(y\). Equation (3) is a Linear Equation of 3 Variables \(x\), \(y\) and \(z\). And equation (4) is a Linear Equation of 4 Variables \(x\), \(y\), \(z\) and \(w\).
   
   
2. Following are some Properties of Linear Equations
   1. The co-efficient of atleast one variable must be non-zero
   2. No variable can have power more than 1.
   3. No term can contain product of 2 or more variables.
3. In \(N\) Dimensional Space (where \(N \geq 2\)), Linear Equations of \(K\) Variables (where \(K \leq N\)) are used to represent Linear Coordinate Geometric Objects
   
   In \(2\)-Dimensional Space/Plane, a Linear Equation of 1 or 2 Variables represents a Line.
   
   In \(3\)-Dimensional Space, a Linear Equation of 1, 2 or 3 Variables represents a Plane.
   
   In \(N\)-Dimensional Space (where \(N \geq 4\)), a Linear Equation of 1, 2, 3, ... or \(N\) Variables represents a Hyper-Plane.