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Math Help - solving a system of linear equations

  1. #1
    Junior Member surffan's Avatar
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    solving a system of linear equations

    Solve the system of linear equations and interpret your solution geometrically:
    [1] 2x + y + 2z - 4 = 0
    [2] x - y - z - 2 = 0
    [3] x + 2y - 6z - 12 = 0

    I keep getting really odd numbers with a lot of fractions whenever i try to solve for it.
    First i eliminated x
    [1] 2x+y+2z-4=0 [2] x-y-z-2=0
    [2] - 2x-2y-2z-4=0 [3] - x=2y-6z-12=0
    [4] 3y+4z=0 [5] -3y+5z+10=0
    then i eliminated y to solve for z
    [4] 3y+4z=0
    [5] + -3y+5z+10 = 0
    9z+10=0
    z=-10/9
    then i solved for y
    [4]3y+4(-10/9)=0
    y=1.48
    then i solved for x
    [1]2x+1.48+2(-10/9)=0
    x= 0.371

    so im guessing this means my point of intersection (0.371,1.48, -1.11)
    im not sure if this is correct or if i missed anything, if anyone can give me their input i would really appreciate it. I'm not sure how to interpret it geometrically either.
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  2. #2
    Senior Member
    Joined
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    Your solution is almost correct, except in the last part, you dropped the "-4" term from equation [1] when solving for x.

    The solution is \left(\tfrac{64}{27},\tfrac{40}{27},-\tfrac{10}{9}\right) \approx(2.37,1.48,-1.11).

    The important thing to take from this geometrically is that each equation is a line in 3-dimensional space. Therefore, the solution set to the system of equations represents all the points that are on all three lines simultaneously. Since you found that there is only a single solution to the system of equations, it means that all three lines intersect at a single point.
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