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Parallel and Perpendicular Lines Equations


A Lines can either be intersecting or parallel by straight geometry. Straight lines are said to be perpendicular if product of their respective slopes of their is -1. two perpendicular line can be intersecting too. As we know that two Linear Equations can intersect at just one Point, This is true for two Perpendicular Lines also. Straight lines are define as to be parallel if the slope the theses line are same, considered vertically or horizontally. Till now there is no solution for two Parallel Lines. The reason is These lines do not intersect each other. But the solution for two perpendicular lines can be found by solving linear equations in usual way

What is a Perpendicular Line 

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Now we will see what are Perpendicular Lines, we will simply say that the lines which intersect each other at the angle of 90 degrees are called the pair of perpendicular lines. If the two lines are perpendicular to each other, then they must intersect each other at the angle of 90 degrees. When we write the English alphabets, L, T, H, E etc, we observe that these alphabets are formed by joining the perpendicular lines.

What is Parallel Lines

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A line is the Set of points which extend endlessly in both the directions. We say that a line has no fixed length and so it cannot be drawn on a plane. We say that a pair of lines is parallel line if they are at equal distance at all the points. A very common example of parallel lines is the railway track. The opposite edges of the rectangular table, the opposite edges of a book or notebook all are the examples of a pair of parallel lines Symbol for Parallel Lines "||" is the symbol for parallel lines..

Parallel Lines Equations  

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Line equation is written in this form $y = mx + b$, 

$m$ = The slope of the line
$b$ =  $y$ -intercept. 

As we know from above the $2$ lines are said to be parallel if their y- intercepts are different and slopes are equal.

Let an equation of a line is $y = 5x - 10$

Two lines are said to be parallel if there slopes are equal. 

If the line $y = m_{1} x + b_{1}$, $m_{1} \neq 0$ and $y = m_{2} x + b_{2}$, $m_{2} \neq 0$ are parallel, 

then, $m_{1} = m_{2}$.

Perpendicular Lines Equations  

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Let a equation of a straight line y = mx + b. Where m is we know from above two line are said to be perpendicular if they meet at right angles. Also two line equations are perpendicular when the product of their slope is equal to  -1. 

Here we have two lines 

$y = m_{1}x + b_{1}$, 

$m_{1} \neq 0$ 


$y = m_{2}x + b_{2}$,

$m_{2} \neq 0$ 

So  these lines will be perpendicular, if $m_{2}$ = $\frac{-1}{m_{1}}$

Again two lines gradients are

$m_{1} m_{1}$ and $m_{2}m_{2}$ 

So if $m_{1}m_{1} * m_{2}m_{2} = -1$ then lines are perpendicular or vice versa. 

Here let's understand with an example

The standard form of the equation: $ax + by = c$

by flipping a and b and changing a sign we have 

$\geq - bx + ay = c$ (or any value) is the perpendicular equation to the $ax + by = c$.


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Some of the solved examples of parallel and perpendicular line equations are given below, 
Example 1:

Equation $12x + 13y = 21$, passing through the point $(1, 5)$, find the perpendicular equation line?


Given equation is $12x + 13y = 21$

Find the slope = $\frac{−coeff.\ of\ x}{coeff.\ of\ y}$ = $\frac{−12}{13}$

As we know the slopes of perpendicular lines are negative reciprocal to each other.

Perpendicular line slope =  $\frac{13}{12}$

Now, Perpendicular line equation is ,

$y - 5$ = $\frac{13}{12}$ $(x - 1)$

$12(y - 5) = 13(x - 1)$

$12y - 60 = 13x - 13$

$13x - 12y + 47 = 0$

Which is passing through $(1,5)$
Example 2:

Determine $y = 2x - 5$ and $-7x + y = 4$ are parallel.


Given lines are $y = 2x - 5$ and $-7x + y = 4$

Slope intercept form of the given lines is

$y = 2x - 5$  

Slope $(m_{1}) = 2$

$y = 7x + 4$ 

Slope $(m_{2}) =7$

$m_{1} \neq m_{2}$

Since the slopes of the above two lines are not equal, the two lines are not parallel.