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System of Linear Equations - Proving

  1. Oct 3, 2011 #1
    1. The problem statement, all variables and given/known data

    Given that the values for a, b, c, d, e and f for the system ax+by=e, cx+dy=f has two different solutions. Show that ax+by=0, cx+dy=0 also has two different solutions.

    2. Relevant equations

    3. The attempt at a solution

    There're three cases of how two straight lines can intersect:
    (i) At only 1 point: unique solution
    (ii) They are parallel and do not intersect: no solution
    (iii) They are the same line: infinitely many solutions

    I assume by two, it means infinitely many solutions?

    Hence, the matrix [a b; c d] is singular? I think I need to use row equivalent or equivalent matrix for this?

    Last edited: Oct 3, 2011
  2. jcsd
  3. Oct 3, 2011 #2


    User Avatar
    Homework Helper

    Hint: Think about what it would mean for [ax + by = 0, cx + dy = 0] to have no more than 1 solution.
  4. Oct 3, 2011 #3


    Staff: Mentor

    Yes, because two straight lines can't intersect in exactly two points.
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