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Math Help - Linear Algebra System Basic Proof

  1. #1
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    Linear Algebra System Basic Proof

    Given a system of the form

    mx + y = c
    nx + y = d

    where c, d, m and n are constants:

    A. Show that the system will have a unique solution if m does not equal n.
    B. If m = n, show the system will only be consistent if c = d.
    C. Give a geometric interpretation of parts A and B

    I have no idea where to begin with that. I tried using elementary row operations on the augmented matrix, but I couldn't gather anything from it and I fear I may have done it incorrectly.

    Any help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by WTFsandwich View Post
    Given a system of the form

    mx + y = c
    nx + y = d

    where c, d, m and n are constants:

    A. Show that the system will have a unique solution if m does not equal n.
    B. If m = n, show the system will only be consistent if c = d.
    C. Give a geometric interpretation of parts A and B

    I have no idea where to begin with that. I tried using elementary row operations on the augmented matrix, but I couldn't gather anything from it and I fear I may have done it incorrectly.

    Any help would be greatly appreciated.
    A. These are linear functions.

    You should know that you can express a linear function in the form y = ax + b, with a as the gradient and b as the y intercept.


    Rearranging the equations gives:

    y = -mx + c
    y = - nx + d.

    If m = n then the gradients will be the same and the lines will be parallel. Therefore there will not be a solution.

    For m \neq n we have

    -mx + c = -nx + d

    c - d = (m - n)x

    x = \frac{c - d}{m - n}.

    Substituting back into one or both of the equations, you will see there is only one solution.


    B. If m = n, like I said, the lines are parallel.

    However if c = d as well, the equations are in fact, identically equal. In other words, they are the same line.


    C. Done in A and B - not sure if it's what you wanted...
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