Combining Parametric Equations

[robbondo]

Homework Statement

Show that every point on the line v = (1,-1,2) + t(2,3,1) satisfies the equation
5x - 3y - z - 6 = 0

Homework Equations

The Attempt at a Solution

So what I did was solve the equation v by adding the x,z,and z components to get

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

So I'm thinking that if I can combine these into one equation that I would end up getting the answer. Problem is I can't figure out how to combine the three equations in an equation with all three variables. I keep getting the function in terms of two variables, or the wrong answer all together. One method I used was solving the z equation for t and then plugging it into the x equation and then the y equation. I tried setting they both equal to zero...

t = z -2
y = 3z-7 y - 3z + 7 = 0
x = 2z-3 x - 2z + 3 = 0

y - 3z + 7 = x -2z + 3

y - z - x + 4 = 0

Obviously this isn't the correct answer. What am I doin' wrong here?

 

[Dick]

Just put your parametric t expressions for x,y and z into 5x-3y-z-6. Do you get zero?

 

[robbondo]

Cool... that works. Why didn't my method work also? Is there more than one equation that can solve that parametric equation?

 

[Mystic998]

Stupid answer goes here.

 

[robbondo]

Was my algebraic logic correct though? The answer I got was not off by a constant.

 

[Mystic998]

Never mind, I'm an idiot. A line doesn't determine a plane. Wow. Your work is fine, it's just you found a plane that the line is in that isn't the plane you started with.