Parallel Lines Shortcut

Parallel lines in Analytical Geometry are two lines with the same slope. Basically, to solve for the equation of the line through a point and parallel to a given line with a known slope can be done in series of steps. However, using a shortcut formula, it can be done by one amazing move.

Parallel lines1


Given a line ax+by+c=0, find the equation of the line through  (x_1,y_1).

The shortcut can be made if we observe a pattern when solving. Finding the line with a given conditions above has a pattern. Thus we can create a shortcut formula.


Step 1: Find the slope of the given line





From here, the slope of the given line is  \displaystyle\frac{-a}{b}

Using the point-slope form, we can now solve for the equation of the required line.


But  m=\displaystyle\frac{-a}{b}





Worked Problem 1:

Find the equation of the line through  (4,-2)  and parallel to  2x-y=4.


Using the derived formula above we have,





Worked Problem 2:

Find the equation of the line through  (5,4)  and parallel to the line  2x+9y=4.


Using the derived formula we have,





Worked Problem 3:

ABCD  is a rectangle with coordinates  A(1,3),B(3,1),C(6,4).  Find the line containing the side  \overline{AD}.


By drafting the figure as follows we have

paralle lines 2

We don’t have the coordinate of D but we can solve for the equation of the line \overline{BC}

Solving for slope of  \overline{BC},






To solve for the equation of the line BC use either point B or C and use the point slope form.





We can now find the equation of the line AD using the shortcut formula above using point A and the line BC.



Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

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