Line

A line is a one dimensional representation which has no thickness and which can be extended to infinite length.




Standard form a line

The equation Ax + By = C, where A, B, C are constants and A, B both together not equal to zero is called the standard form of a line.
Example: - 3x + 5y = 2 is an example of a line in the standard form.
There will be no fractions in the standard form of a line.

Slope of the line

To find the slope of the line in the standard form, we use the formula
2

Lets consider a few examples

Example 1: -
Find the equation of a line passing through the point (3, 4) with slope 2 in the standard form.
Solution: -

Given that the required line is passing through the point (3, 4).
Slope of the line = 2.
First lets write the line using point slope form.
We know that when a point (x1, y1) and slope m is given, then the point slope form is
y - y1 = m (x - x1).
Substituting the slope and points we get
y - 4 = 2 (x - 3)
Expanding we get
y - 4 = 2x - 6
Now we to take all variables to one side, keeping the constants on the other side.
Adding 4 on both sides, we get
y - 4 = 2x - 6
   +4         + 4

y     =  2x  - 2
Now we subtract 2x from both sides
   y     = 2x - 2
   -2x     -2x      
y -2x =           -2
So the required equation is -2x + y = -2

Example 2: -
Find the slope of the equation 3x - 4y = 9.
Solution: -
The given equation is in the standard form.
We know that the formula for finding slope is given by 2
Here coefficient of x = 3 and coefficient of y = -4
So slope of the equation = -(3/(-4)) = 3/4

Example 3: -
Find the equation of the line passing through the points (4, 2) and (5, 3) and hence find the slope.
Solution: -

Given that the required line is passing through the points (4, 2) and (5,3).
First lets write the line using two point form.
We know that when two points (x1, y1) and (x2, y2) are given, then the equation of the line is  is
y - y1 = m (x - x1) where m = (y2 - y1)/(x2 - x1)
Here (x1, y1) = (4, 2) and (x2, y2) = (5, 3)
So m = (y2 - y1)/(x2 - x1) = (3 - 2)/(5 - 4) = 1/1 = 1
Substituting the slope and points we get
y - 2 = 1 (x - 4)
Expanding we get
y - 2 = x - 4
Now we to take all variables to one side, keeping the constants on the other side.
Adding 2 on both sides, we get
y - 2 = x - 4
   +2      + 2

y     =  x  - 2
Now we subtract x from both sides
   y     = x - 2
   -x      -x      
y -x =           -2
So the required equation is -x + y = -2
We know that the formula for finding slope is given by 2
Here coefficient of x = -1 and coefficient of y = 1
So slope of the equation = -(-1/(1)) = 1

Example 4: -
Find the equation of the line passing through the points (-3, 4) and (-7, 9) and hence find the slope.
Solution: -

Given that the required line is passing through the points (-3, 4) and (-7, 9).
First lets write the line using two point form.
We know that when two points (x1, y1) and (x2, y2) are given, then the equation of the line is  is
y - y1 = m (x - x1) where m = (y2 - y1)/(x2 - x1)
Here (x1, y1) = (-3, 4) and (x2, y2) = (-7, 9)
So m = (y2 - y1)/(x2 - x1) = (9 - 4)/(-7 - (-3)) = 5/-4 = -5/4
Substituting the slope and points we get
y - 4 = -5/4 (x + 3)
Expanding we get
y - 4 = -5/4x - 15/4
Here we have a fraction in the equation. So we multiply the whole equation by the denominator 4.
We get
4y - 16 = -5x  -15.
Now we to take all variables to one side, keeping the constants on the other side.
Adding 16 on both sides, we get
4y - 16 = -5x - 15
     +16          + 16

  4y     = -5x  + 1
Now we subtract -5x from both sides.  That is add 5x on both sides
   4y     = -5x + 1
  +5x       + 5x      
4y + 5x =         1
So the required equation is 5x + 4y = 1
We know that the formula for finding slope is given by 2
Here coefficient of x = 5 and coefficient of y = 4
So slope of the equation = -(5/(4)) = -5/4

Try Yourself: -

A line is passing through (2,3) and (4,2).  Write the equation in the standard form.  Also find the slope of the line.

Answer: -
Equation x + 2y = 8 slope = -1/2