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Math Help - coordinate

  1. #1
    Senior Member sfspitfire23's Avatar
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    coordinate

    In a coordinate system, if three points (5, 3), (x, 4) and (3, 2) lie on a same line, then find a value of x?


    Now, the slope appears to be 1/2 from the two given points.

    Does this mean that x will be a number than when used to calculate the slope of the line with any of the other two points will still be 1/2? In this case, 7? Or, do we have to develop the equation of the line and substitute (x,4) into the equation of the line in which case I get 5.
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  2. #2
    A Plied Mathematician
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    They should both give you the same answer. Can you show a bit more work?
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  3. #3
    MHF Contributor Unknown008's Avatar
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    You should get the same answer from both methods, because they are somewhat related. The formula I use for finding the gradient is:

    \frac{y_2 - y_1}{x_2 - x_1} = m

    Then, I would use:

    \frac{y - y_1}{x - x_1} = m

    To get the equation of the line. But, since both equals to m, I can directly say that:

    \frac{y_2 - y_1}{x_2 - x_1} = \frac{y - y_1}{x - x_1}

    So, it's the same as making both gradients equal.

    A shorter way that I came to.

    Rearrange your coordinates so that the y coordinates are in ascending order.

    (3, 2) , (5, 3) , (x , 4)

    Looking at the y coordinates, it increases by 1 each time. And since all the points lie on a straight line, you can conclude that (5, 3) is the midpoint of (3, 2) and (x, 4). In which case, you will have an arithmetic sequence for the x coordinate too. It's 3, 5 then x. Since it increases by 2, you add 2 to 5 to give 7, the value of x.

    In any case, you should get x = 7.
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  4. #4
    Senior Member sfspitfire23's Avatar
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    Wow nevermind. Simple arithmetic kills :-p
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  5. #5
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    Hello, sfspitfire23!

    Did you make a sketch?

    I did . . . and baby-talked my way to the answer.



    In a coordinate system, points A(5,3),\;B(x, 4),\;C(3, 2) are collinear.

    Find a value of x.
    Code:
          |                                       B
          |                                  (x,4)*
          |                                   *   :
          |                               *       :1
          |                       A   *           :
          |                  (5,3)o - - - - - - - + 
          |                   *   :       ?
          |               *       :1
          |       c   *           :
          |  (3,2)o - - - - - - - +
          |               2
          |
      - - + - - - + - - - - - - - + - - - - - - - +
          |       3               5

    Going from C to A, we move 2 units right and 1 unit up.

    Going from A to B, we move ? units right and 1 unit up.

    It is obvious (I hope) that: . ? \,=\,2

    Therefore: . B(7,4)\,\text{ and }\,x = 7

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