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Math Help - Parallel Proof

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    Parallel Proof

    Prove that nonvertical lines are parallel if and only if their slopes are equal.
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    Lines can be written in the form \displaystyle y = mx + c, where \displaystyle m is the gradient and \displaystyle c is the \displaystyle y intercept.

    So we could write the equation of two lines as

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

    \displaystyle m_1x - m_2x = c_2 - c_1

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

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


    Obviously there is no solution when the denominator is \displaystyle 0, in other words, where \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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