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Math Help - Confused about equation of a line

  1. #1
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    Confused about equation of a line

    Equation says this:
    \frac{x-3}{-2}=\frac{y+1}{-1}=\frac{z}{3}

    But the book tells me it can also be written like this:
    d: \left\{\begin{matrix}<br />
x-2y-5=0\\3y+z+3=0 <br />
\end{matrix}\right.

    I got confused on how that could be so I tried to get the second expression of a line from the first expression of the line by trying cross multyplication:
    \frac{x-3}{-2}=\frac{y+1}{-1} =\: \:  -1(x-3)=-2(y+1)

    But I get this which is close but not it:
    -x+2y+5=0

    Although when I do this:
    \frac{y+1}{-1}=\frac{z}{3} =\: \:  -z=3(y+1)

    I get something that resembles the equation in the second part:
    3y+z+3=0

    Am I doing something wrong here? How does this form a line exactly because I grew up with the y = mx+b equation and this is the first time I am seeing this.
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  2. #2
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    Quote Originally Posted by Bman900 View Post
    Equation says this:
    \frac{x-3}{-2}=\frac{y+1}{-1}=\frac{z}{3}
    But the book tells me it can also be written like this:
    d: \left\{\begin{matrix}<br />
x-2y-5=0\\3y+z+3=0 <br />
\end{matrix}\right.
    What the text calls d is actually two planes.
    The given line is the intersection of those two planes.
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  3. #3
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    Quote Originally Posted by Bman900 View Post
    Equation says this:
    \frac{x-3}{-2}=\frac{y+1}{-1}=\frac{z}{3}

    But the book tells me it can also be written like this:
    d: \left\{\begin{matrix}<br />
x-2y-5=0\\3y+z+3=0 <br />
\end{matrix}\right.

    I got confused on how that could be so I tried to get the second expression of a line from the first expression of the line by trying cross multyplication:
    \frac{x-3}{-2}=\frac{y+1}{-1} =\: \:  -1(x-3)=-2(y+1)

    But I get this which is close but not it:
    -x+2y+5=0
    It is exactly if you multiply the equation by -1.

    Although when I do this:
    \frac{y+1}{-1}=\frac{z}{3} =\: \:  -z=3(y+1)

    I get something that resembles the equation in the second part:
    3y+z+3=0
    Yes, that is how they got those equations.

    Am I doing something wrong here? How does this form a line exactly because I grew up with the y = mx+b equation and this is the first time I am seeing this.
    y= mx+ b is the equation of a line in the plane. In three dimensions, you need 2 equations to specify a 3- 2= 1 dimensional line. A single equation in three variables determines a 3- 1= 2 dimensional surface.

    You can break
    \frac{x- a}{A}= \frac{y- b}{B}= \frac{z- c}{C}
    into two independent equations just as you did. (There are other ways to form new equations but they are not independent equations.)
    Last edited by HallsofIvy; December 29th 2010 at 01:44 AM.
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  4. #4
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    Ah I see it now, I was thinking about multiplying it by -1 but I was wondering why didn't they just leave it the way it comes out of cross multiplication. Anyway thank you for the help!
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