Equations of straight line

A line-segment is obtained by joining two points in space. However, the two points may be in the same plane and we are interested in a line which is in a two dimension planes.
An infinite number of lines can be drawn through a given point as shown below 

 But we get only one possible line through two given points. A line through two points in a plane is unique. 

We see that line in the picture above passes through two points $(x_1, x_2)$ and $(y_1, y_2)$. These are the coordinates of two points. We see that there is some relation between x and y for every point on the line. We find this relation and name it as equation of line.
Equation of line may be in any form as below 

• Point-slope form 
• Slope-intercept form 
• Standard form 
• General form  

Before discussing any form of equation of line, we will define slope of the line.
Slope of a line:  It is defined as the ratio of change in y for a unit change in x. It means for a unit increase in x, what we get as the change in y is called the slope of the line.
In simple words we defined the slope of a line is the ratio of rise to run. We denote slope by small letter m.
Mathematically, we can write it as 

                                                            $Slope, \, \, m=\dfrac{rise}{run}$

For the line shown in picture above, the first point is $(x_1, y_1)$ and second point is $(x_2, y_2)$ , 

We  also observer that rise is from $y_1$ to $y_2$ thus $rise=y_2-y_1$

and run is from $x_1$ to $x_2$ thus $run=x_2-x_1$

Therefore, slope of the line is  $Slope, \, \, m=\dfrac{rise}{run}=\dfrac{y_2-y_1}{x_2-x_1}$

Let's take an example. 

Find the slope of line passing through the points (2,5) and (4,11)

Solution: Here, first point is (2,5) and second point is (4,11)

Therefore, $x_1=2, y_1=5$ and $x_2=4 , y_2=11$

Plug in the given values in above formula, 

$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{11-5}{4-2}=\dfrac{6}{2}=3$

Thus, slope of the line passing through points (2, 5) and (4, 11) is 3

We save m=3

Using the above formula for the slope, we can find the slope of any line passing through the given two points.

Now we will discuss the point slope of the equation of a line.

Point-Slope form of equation of line:

Consider the line in the picture below 

We see that there is third point on the line (x, y), It's a general point which can be anywhere on the line.
Using the points $(x_1, y_1)$ and $(x, y)$, 
we get ,                                  slope $m=\dfrac{y-y_1}{x-x_1}$

Cross-Multiplying, we get 

                                           $m(x-x_1)=(y-y_1)$

We can write it as                     $y-y_1=m(x-x_1$

It's called the point slope form of the equation of line. 

Suppose point is (h, k) and slope of the line is m 

Then, point slope form is               $y-k=m(x-h)$

Let's take an example. Find the equation of line in point slope form for the line passing through the point (5, 3) and having slope 2. 

Solution: Here the point is (5,3). compare it with (h, k) we get
h=5 and k=3 and given slope is m=2

Plug in the given values in the formula for point-slope form, 

                                                        $y-3=2(x-5)$

It's the point slope form.

Slope-intercept form of the line: 

An equation of the line in the form of $y=mx+b$ is known as slope-intercept of the line. 

We have point-slope form as           $y-k=m(x-h)$

Add k both sides, we have                  $y=m(x-h)+k $ 

Simplify                                              $y=mx-mh+k$

                                                            $y=mx+(k-mh)$

Plug in b for k-mh we get 

                                                            $y=mx+b$

Here b is known as the y-intercept and b=k-mh and (h, k) is the point through which line passes. 

And (0, b) is the y-intercept in point form. 

Let's take an example.
Find the equation of line in slope intercept form passing through the (0, 5) and having slope 3. 
Solution:  Here slope of the line, m=3 and y-intercept is (0, 5). Just compare (0, 5) with (0, b), we get b=5
Now plug in m=3 and b=5 in the slope intercept equation y=mx+b, we get
                                                           $y=3x+5$
It's the answer to the above problem. 

General equation of line:  General equation of line is in the form of $ax+by+c=0$.
Let's reduce above equation $y=3x+5$ into general form.
Subtract y from both sides, we have $0=3x+5-y$
Or we can write it as               $3x-y+5=0$
It's the general form of the equation of line. 
Standard form: Standard equation of line is in the form $of ax+by=c$
Let's reduce above equation of line into the standard form. 
We have  $3x-y+5=0$
Subtract 5 from both sides, 
                    $3x-y=-5$
It's the standard equation of line.
Note: 

• The slope of two parallel lines is equal i.e. two parallel lines have same slope. 
• The product of the slope of two perpendicular line is -1. 

Suppose two lines have slopes $m_1$ and $m_2$ then according to the above note, we have
If $m_1 \cdot m_2=-1$, then the two lines are perpendicular.
and if $m_1=m_2$, then the two lines are parallel. 

Now try these problems. 

1.  Find the slope of line passing through the points (3, 5) and (4, 7)
2. Find the slope -intercept form of equation of line passing through (-3,3) and parallel to the line $3x+2y=1$
3. Find the standard equation of line passing through point (-2, -3) and perpendicular to the line $y=4x-3$