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Thread: Parallel Proof

  1. #1
    Mar 2010

    Parallel Proof

    Prove that nonvertical lines are parallel if and only if their slopes are equal.
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  2. #2
    MHF Contributor
    Prove It's Avatar
    Aug 2008
    Lines can be written in the form $\displaystyle \displaystyle y = mx + c$, where $\displaystyle \displaystyle m$ is the gradient and $\displaystyle \displaystyle c$ is the $\displaystyle \displaystyle y$ intercept.

    So we could write the equation of two lines as

    $\displaystyle \displaystyle y=m_1x + c_1$ and $\displaystyle \displaystyle y=m_2x + c_2$.

    You should know that parallel lines never cross. These two lines will cross when their equations are equal. So

    $\displaystyle \displaystyle m_1x + c_1 = m_2x + c_2$

    $\displaystyle \displaystyle m_1x - m_2x = c_2 - c_1$

    $\displaystyle \displaystyle x(m_1-m_2) = c_2 - c_1$

    $\displaystyle \displaystyle x = \frac{c_2 - c_1}{m_1-m_2}$.

    Obviously there is no solution when the denominator is $\displaystyle \displaystyle 0$, in other words, where $\displaystyle \displaystyle m_1 = m_2$.

    So the only time when there will not be a point of intersection is when the two gradients are equal.

    In other words, the lines are parallel when their gradients are equal.
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