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Math Help - Linear algebra question...

  1. #1
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    Linear algebra question...

    Given a system of the form

    -m1x1 + x2 = b1
    -m2x2 + x2 = b2

    where m1, m2, b1, and b2 are constants:

    (a) Show that the system will have a unique solution if m1 != m2
    (b) If m1 = m2, show that the system will be consistent only if b1 = b2
    (c) Give a geometric interpretation to parts (a) and (b)

    Not sure where to even begin with this, thanks in advance!
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  2. #2
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    Quote Originally Posted by pakman View Post
    Given a system of the form

    -m1x1 + x2 = b1
    -m2x2 + x2 = b2

    where m1, m2, b1, and b2 are constants:

    (a) Show that the system will have a unique solution if m1 != m2
    (b) If m1 = m2, show that the system will be consistent only if b1 = b2
    (c) Give a geometric interpretation to parts (a) and (b)

    Not sure where to even begin with this, thanks in advance!
    Hello,

    first I assume that there is a typo:

    \begin{array}{l}-m_1 x_1 + x_2 = b_1 \\-m_2 x_1 + x_2 = b_2\end{array} . Subtract columnwise(?):

    -m_1x_1 + m_2 x_1 = b_1 - b_2~\iff~x_1 = \frac{b_1-b_2}{m_2-m_1} . As you easily can see there exists only a solution for x_1 if m_1 \ne m_2

    If m_1 = m_2 then the 2 lines are parallel and there doesn't exist a common point, that means there doesn't exist a solution for x_1

    If m_1 = m_2~\wedge~b_1 = b_2 the lines are equal that means there are infinite many common points.
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