Answer:
The equation 
represents in R^2 c) a line
The equation represents in R^3 a) a plane
The equation 
represents in R^3 a) a plane
The equation 
represents d) a plane
The pair of equations 
represents a) a line
Step-by-step explanation:
Let's start by studying each question :
1)
In R^2 , 
with 
∈ IR is the equation of all vertical lines
In this case, the only free variable is the variable ''y''
R^2 has two dimensions (x and y) so if we set we will have only one free variable in R^2 (which is a line in R^2). Therefore, represents a line in R^2
Now, in R^3 we have three dimensions (x, y and z) so if we set we will have only two free variables (y and z) and will represent a plane (which have two dimensions) in R^3.
2) The equation in R^3 has the free variable ''z'' and given that we select a value (for example for ''x'') the another value from the variable ''y'' is determined. Finally, we have the free variables ''z'' and ''x'', and the variable ''y'' restricted for our choice of the variable ''x''.
The equation in R^3 (given that we have two free variables) represents a plane.
Using the same reasoning, the equation represents a plane (given that it has two free variables : ''y'' and ''x'')
Finally, the pair of equations set values for ''y'' and ''z'' leading us ''x'' as the free variable. With only one free variable we will have a ''one dimensional'' geometric form. The one dimensional forms in R^3 are lines.
The final answer is a) a line.
