The equation of a straight line can be written in different forms depending on the data given. The different forms are as follows:
•    SLOPE INTERCEPT FORM OF A LINE
•    POINT SLOPE FORM OF A LINE
•    TWO POINT FORM OF A LINE
•    THE INTERCEPT FORM OF A LINE
•    NORMAL FORM OR PERPENDICULAR FORM OF A LINE
•    DISTANCE FORM OF A LINE

Let’s discuss a few of these in detail.

Slope Intercept Form of a Line

The equation of a line with slope m and making an intercept c on y – axis is y = mx + c

Proof: Let the given line intersects y – axis at B and makes an angle θ with x – axis. Then m = tan θ. Let P (x, y) be any point on the line. Draw PM perpendicular to x – axis and BN perpendicular to PM.

Clearly ∠NBP = θ, BN = OM = x and PN = PM – NM = PM – OB = y – c
From ΔPNB, we have
tan θ = PN/BN = (y – c)/x
m = (y – c)/x
y = mx + c, which is the required equation of the line.

Important Remarks

1. If the line passes through the origin, then 0 = m0 + c; c = 0. Therefore, the equation of a line passing through the origin is y = mx, where m is the slope of the line.

2. If the line is parallel to x – axis, then m = 0, therefore the equation of a line parallel to x – axis is y = c.

Point – Slope Form of a Line

The equation of a line which passes through the point (x1, y1) and has the slope m is
y – y1 = m(x – x1)

Proof: Let Q (x1, y1) be the point through which the line passes and let
P (x, y) be any point on the line. Then, slope of the line is y – y1
                                                                                    x – x1
But, m is the slope of the line.
So, m = y – y1
             x – x1
y – y1 = m(x – x1)
Hence, y – y1 = m(x – x1) is the required equation of the line.

Two – Point Form of a line

The equation of a line passing through two points (x1, y1) and (x2, y2) is
y – y1 = y2 – y1    (x – x1)
             x2 – x1
Proof: Let m be the slope of the line passing through (x1, y1) and (x2, y2). Then,
m = y2 – y1
       x2 – x1
So, the equation of the line is
y – y1 = m (x – x1)   [Using point – slope form]
Substituting the value of m, we obtain
y – y1 = y2 – y1   (x – x1)
             x2 – x1
This is the required equation of the line in two point form.

Now try it yourself!  Should you still need any help, click here to schedule live online session with e Tutor!

About eAge Tutoring :

eAgeTutor.com is the premium online tutoring provider.  Using materials developed by highly qualified educators and leading content developers, a team of top-notch software experts, and a group of passionate educators, eAgeTutor works to ensure the success and satisfaction of all of its students.  

Contact us today to learn more about our guaranteed results and discuss how we can help make the dreams of the student in your life come true!

Reference Links :

• http://en.wikipedia.org/wiki/Linear_equation#General_form
• http://en.wikipedia.org/wiki/Slope
• http://en.wikipedia.org/wiki/Intercept
• http://en.wikipedia.org/wiki/Perpendicular