Linear equation

A linear equation is an  equation representing a line.

Example: -
y = 4x + 2 is an example of a line.


  There are different form of representing a line.They are:-
        1. Standard form
        2. Slope Intercept form
        3. Point slope form
        4. Two point form

Lines are always first degree equations

Standard form of a linear  equation

The standard form of a linear equation is given by Ax+ By = C where A, B and C are constants and x and y are variables.
Example: -
3x + 4y = 7 is an example of a straight line in the standard form.  Here A = 3,       B = 4 and C = 7.

The equation 9x + 7y = 8 is also an example of straight line in the standard form.  Here A = 9, B = 7 and C = 8.

Lets work on a few examples: -

Example 1: -
Find the equation of the line passing through the points (3, 2) and (4, 2) and express it in the standard form.
Solution: -
We know that the line passing through two points (x1, y1) and (x2, y2) is given by
y - y1 = m (x - x1) where m is the slope of the line, which is given by                      m =  (y2 - y1)/(x2 - x1)
Given that the line passes through (x1, y1)  = (3,2) and  (x2, y2) = (4,2).
Lets first find the slope.
Slope m = (y2 - y1)/(x2 - x1) = (2 - 2 )/(4 - 3) = 0/1 =0
Substituting in the equation of the line, we get
y - 2 = 0 (x - 3)
That is y - 2 =0
That is y = 2 is the required line.


In the above problem, A = 0.


Example 2: -
Find the equation of the line passing through the points (4, 9) and (3, 7) and express it in the standard form.
Solution: -
We know that the line passing through two points (x1, y1) and (x2, y2) is given by
y - y1 = m (x - x1) where m is the slope of the line, which is given by                      m =  (y2 - y1)/(x2 - x1)
Given that the line passes through (x1, y1)  = (4, 9) and  (x2, y2) = (3 , 7).
Lets first find the slope.
Slope m = (y2 - y1)/(x2 - x1) = (7 - 9 )/(3 - 4) = -2/-1 = 2
Substituting in the equation of the line, we get
y - 9 = 2 (x - 4)
y - 9 = 2x - 8
Adding 8 on both sides, we get
y - 1 = 2x
subtracting y on both sides we get
2x - y = -1, which is the required equation.  Here A = 2, B = -1and C = -1.

Example 3: -
Find the equation of the line passing through (3, 5) and having a slope 4.
Solution: -
We know that if a line passes through a point (x1, y1) and have slope m, then the equation is given by y - y1 = m (x - x1).  This form is known as point slope form.
Given that the line passing through (3, 5).  That is (x1, y1) = (3, 5).
Slope is 4.  That is m = 4
Therefore the required line is
y - 5 = 4 (x - 3)
y - 5 = 4x - 12
adding 12 both sides, we get
y + 7  = 4x
subtracting y on both sides, we get
4x - y = 7, which is the required equation.

Slope of a segment of a line is the tangent of the inclination of the line with the x axis. Slope of different segments of a line are equal.  Hence slope of a non vertical line is defined as the number which is the slope of every segment of the line.
Slope is also known as gradient.

Try yourself: -
1. Find the equation of the line passing through the points (3,4) and (5,6) and express it in the standard form.
2.  Write the line 2x = y + 9 in the standard form and identify A, B and C.

Answers: -
1. x - y = -1.
2. 2x - y = 9 ; A = 2, B = -1 and C = 9