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

Derivation:

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

ax+by+c=0

by=-ax-c

y=\displaystyle\frac{-ax-c}{b}

y=\displaystyle\frac{-ax}{b}-\displaystyle\frac{c}{b}

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.

y-y_1=m(x-x_1)

But  m=\displaystyle\frac{-a}{b}

y-y_1=\displaystyle\frac{-a}{b}(x-x_1)

by-by_1=-ax+ax_1

ax+by=ax_1+by_1

 

Worked Problem 1:

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

Solution:

Using the derived formula above we have,

ax+by=ax_1+by_1

(2)x+(-1)y=(2)(4)+(-1)(-2)

2x-y=10

2x-y-10=0

Worked Problem 2:

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

Solution:

Using the derived formula we have,

ax+by=ax_1+by_1

(2)x+(9)y=(2)(5)+(9)(4)

2x+9y=46

2x+9y-46=0

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}.

Solution:

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},

m_{BC}=\displaystyle\frac{y_2-y_1}{x_2-x_1}

m_{BC}=\displaystyle\frac{4-1}{6-3}

m_{BC}=\displaystyle\frac{3}{3}

m_{BC}=1

 

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

y-y_1=m(x-x_1)

y-1=1(x-3)

y-1=x-3

x-y-2=0

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

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