Equations of Lines and Planes

Question

If the line L is given by x= 2+ 3t , y=1- 2t , z= t, then it could also be written in symmetric form as

(x-2)/6 = (y-1)/ = z/

Answer 1

Solution:

In order to solve this, you have to understand that these are parametric equations where t represents a common solution for the 3 equations or t represent the intersection of the the 3 lines

solve for t in each equation, x = 2 + 3t, add -2 to both sides, x -2 = 3t divide by 3 both sides t = (x-2)/3

y = 1 -2t, add -1 to both sides y -1 = -2t, multiply by -1 both sides , 1-y = 2t

divide both sides by 2, t = ( 1-y )/ 2

z = t , rearrange t = z

therefore t = (x-2)/3 = ( 1-y ) / 2 = z but your answer requires that ( x-2)/6 = (y-1)/ = z/

Then to get ( x-2)/6 you must multiply (x-2)/3 by 1/2 = (x -2) / 3 (1/2) = ( x-2)/6 you then have to multiply all the other equations by 1/2 then( 1-y)/2 (1/2) = ( 1-y)/4 and z(1/2) = z/2 and you get

t = ( x-2)/6 = ( 1-y)/4 = z/2 but the middle term should be ( y-1) NOT ( 1 -y) multiply by -1 top and bottom

(-1)( 1-y)/ -1( 4) = ( y-1)/ -4

The whole set becomes ( x-2)/6 = ( 1-y)/-4 = z/2 and that is the form you are looking for