Parametric Representation of A Solution Set

In describing the Solution Set of a linear equation, we often use parametric representation. 

Solve the Linear Equation: 
1) x+2y=4 

 *to find the solution set of a equation involving two variables, solve for one of the variables in terms of the other variable. 

                                           x=4-2y

 *In this form, the variable y is free which means it can take any real value.The variable x is not free because it's value depends on the value assigned to y. 

 *To represent the infinite number of solutions of this equation, we use a third variable j called a parameter. By letting y=j, you can now represent the solution set. 

            *Let y=j, j is a real number. 
              x=4-2j
              Let j=1
              2y=4-x 
                y=2-1/2x
                x=2
                y=1 
The Solution Set is: (2,1) or (4,2).  

2) 3x+2y-z=3
    3x=3-2y-z
           3
      x=1-2/3y-1/3z

*Let y=s, s is a real number, z=t, t is a real number. 
  x=1-2/3s+1/3t
  x=1-2/3(1)+1/3(1)=2/3
  
The Solution Set is: (2/3,1,1) 

3) 1/2x-1/3y=1 
     (1/2x-1/3y=1)6
       3x-2y=6
       3x=6+2y
              3
         x=2+2y
                   3
*Let y=t, t is any real number. 
  x=2+2/3t
  t=1 
   x=2+2/3t
   x=2+2/3 
   x=8/3

The Solution Set is: (1, 8/3)