How is this considered a line?

Question

x=1+t, y=2t, z=4-3t

Ok I can say t = 1 - x
and plug it in for y = 2t
y = 2(1-x) = 2 - 2x

But I didn't include z so how can this be a line... I'm so confused I really need help.

Answer 1

ok, this can be easy.

Set up a chart with x,y,z, and t... a table of values

Now pick values for t then compute the corresponding values for x,y, and z

example
T= ( X, Y, Z)
1, ( 2 2 1)
2, ( 3, 6, -2)

Ok fill out your chart until you are satisfied with values. These correspond to points in 3d space. Connect the dots to form the line.
I hope this helps

Answer 2

OK I can say t = 1 - x
and plug it in for y = 2t
y = 2(1-x) = 2 - 2x


that works in plane
you must consider also the z-coordinate
I think you need all information
x=1+t, y=2t, z=4-3t is equivalent with
y=2-2x and z=t which means
y=2-2x and z takes any value

You can see that projecting the line on the plane z=0 you get
the line y-2x=0, or you can project it on any plane z=t.

Answer 3

The line exists in 3-dimensional space. To solve, first express each equation in terms of t:

In equation 1:
x = 1 + t, or
1 + t = x
Subtract 1 from both sides:
t = x - 1

In equation 2:
y = 2t, or
2t = y
Divide both sides by 2:
t = y/2

In equation 3:
z = 4 - 3t, or
4 - 3t = z
Subtract 4 from both sides:
-3t = z - 4
Multiply both sides by-1:
3t = -z + 4, or
3t = 4 - z
Divide both sides by 3:
3t / 3 = (4 - z) / 3
t = (4-z) / 3

In equation 1, set x = 0:
t = x - 1
t = 0 - 1
t = -1

In equation 2, substitute t with -1 (when x = 0, t = -1):
y = 2t
y = 2(-1)
y = -2

In equation 3, substitute t with -1:
z = 4 - 3t
z = 4 - 3(-1)
z = 4 + 3
z = 7

The first point on the line is (0, -2, 7).

In equation 2, set y = 0;
y = 2t, or
2t = y
2t = 0, 2 not = 0, therefore:
t = 0

In equation 1, substitute t with 0 when y = 0, t = 0):
t = x - 1, or
x - 1 = t
x - 1 = 0
Add 1 to both sides:
x + 1 -1 = 0 + 1
x = 1

In equation 3, substitue t with 0:
z = 4 - 3t
z = 4 - 3(0)
z = 4

The second point on the line is (1, 0, 4)
Connect the two points and you have your line.

Answer 4

It is a line in three space. Equations for lines in three space are different than those for lines in a plane.

In a plane, you have Ax + By + C = 0

In space, you have (x - x0)/a = (y - y0)/b = (z - z0)/c as one of many forms

Note that your x, y, and z values have slopes 1, 2, and -3.

The form you give is called parametric form.

Answer 5

Then, include it:
z = 4-3(1-x) = 1 + 3x
So, as x increases, y decreases linearly, and z increases linearly.
Thus, this is a line.

Answer 6

This is a parametric equation. Each value of t you can pick gives you a distinct x, y, and z value. If you plotted each of these (x, y, z) values as points in 3D space, the points would form a straight line in 3D space.