Question

1) What does the equation x = 4 represent in R^2? a) a circle.b) a plane.c) a line.d) a pointWhat does it represent in R^3? a) a plane.b) a line.c) a circle.d) a point.2) What does the equation y + 3x = 2 represent in R^3? a) a plane.b) a circle.c) a line.d) a point.What does z - 4y = 8 represent? a) a line.b) a circle.c) a point.d) a plane.What does the pair of equations y = 2, z = 8 represent? In other words, describe the set or points (x, y, z) such that y = 2 and z = 8. a) a line.b) a circle.c) a plane.d) a point.

Answer 1

The equation
`x=4` represents in R^2 c) a line

The equation
`x=4` represents in R^3 a) a plane

The equation
`y+3x=2` represents in R^3 a) a plane

The equation
`z-4y=8` represents d) a plane

The pair of equations
`y=2,z=8` represents a) a line

Explanation:

Let's start by studying each question :

1)

In R^2 ,
`x=a` with
`a` ∈ 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
`x=4` we will have only one free variable in R^2 (which is a line in R^2). Therefore,
`x=4` represents a line in R^2

Now, in R^3 we have three dimensions (x, y and z) so if we set
`x=4` we will have only two free variables (y and z) and
`x=4` will represent a plane (which have two dimensions) in R^3.

2) The equation
`y+3x=2` 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
`y+3x=2` in R^3 (given that we have two free variables) represents a plane.

Using the same reasoning, the equation
`z-4y=8` represents a plane (given that it has two free variables : ''y'' and ''x'')

Finally, the pair of equations
`y=2,z=8` 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.