Equation Of A Line

Equation of a line is the equation satisfied by all the points which lie on the line.

 So, we can call every point on the line as a solution of the equation.

 As there are infinitely many points on a line, the equation of a line has infinitely many solutions.

 For every value of x, you can find a value for y by substituting the x value in the equation of the line.

 That is, consider the line whose equation is 2x + y = 4.

 We can find the value of y, when x equals 2.

 Substituting x = 2 in 2x + y = 3, we have

 2(2) + y = 3

 ? 4 + y = 3

 ? y = -1

 So, (2, -1) is a point on the line 2x + y = 4.

 There are various approaches to derive equation of a line.
Point Slope Form of Equation of a Line

We can draw a line if we know its slope and a point through which it passes. And the line we get is unique. Is it? Try.

 Hence we can find the equation of a line when its slope m and a point P(x1, y1) through which passes are given.

 The equation of the line is given as (y – y1) = m(x – x1)
Slope Intercept Form of Equation of a Line :

Suppose if c is the y-intercept of the line, then (0, c) is a point on the line.

 So, the equation (y – y1) = m(x – x1) can be written as (y – c) = m(x – 0)

 That is, y = mx + c

 So, we can find the equation of the line if its slope and y-intercept are known.
Equation of a Line Passing through Two Given Points:

Equation of a  line passing through two given points is also unique.

 So, we can find the equation of a line, if we know any two points on it.

 Suppose line l passes through the points A(x1, y1) and B(x2, y2), then slope of the line l is

 m = y2 - y1 / x2 -x1

Now we know the slope and a point on the line (considering either A or B) and hence we can find the equation of the line using the formula (y – y1) = m(x – x1).

So, the equation of the line is

(y - y1) = y2 - y1 / x2 -x1  (x - x1)

OR

(y - y2) = y2 - y1 / x2 -x1  (x - x2)

That is y - y1/ x - x1 = y2 - y1 / x2 -x1

In general, we can rewrite the equation of a line in the form ax + by + c = 0, where a, b and c are real numbers.