# Equation of a Line

The equation of a line is a linear relation between two variables *x *and *y*, which is satisfied by the coordinates of each & every point on the line and not by any other point.

There are various forms of Equation of a line.

*Example:*

- 2
*x*+ 3*y*+ 7 = 0 which can also be written as 2*x*+ 3*y*= (-7). - 4
*x*+ 3 = 7 where*B*= 0. - 3
*y*+ 2 = 5 where*A*= 0.

**Point Slope Form**

Equation of a line with slope or gradient ** m** and passing through the point (,) is given by

(Â –Â ) =Â Â (Â –Â )

*Example:*

- Equation of line passing from origin (0, 0) and slope 1 is (
*y â€“*0) = (*x*â€“ 0) or*y = x.* - Equation of line passing from (1, 3) and slope 4 is (
*y â€“*3) = 4(*x*â€“ 1) or*y â€“*3 = 4*x*â€“ 4

Hence the line is given by the equation 4*x â€“ y *â€“ 1 = 0.

**Slope Intercept form**

The length of the intercept made by of a line on the *Y-*axis is called* its y*

**-intercept**.

Equation of a line with slope or gradient ** m** and making an y intercept of

*c**units is given*

*by* *Â Â Â Â Â Â Â Â Â Â **Â Â Â Â Â Â Â Â Â Â Â (y – c) = m ( xÂ – 0 )*

i.e Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **Â Â Â Â Â Â Â y=mx + c**

*Example:*

- Equation of line with slope,
*m*= Â & Â y-intercept*c*= 7 is*y =**x*+ 7. - Equation of line with slope,
*m*= 11 & y-intercept*c*= -2 is*y =*11*x*â€“ 2.

- Write the equation of the line 3
*x*= 4*y*+ 7 in general form. - Convert the equation 3
*x*= 4*y*+ 2 in slope intercept form. - Find the equation of the line passing from origin (0, 0) and slope 4.
- Find the equation of the line passing from origin (1, 5) and slope -7.
- Find the equation of the line with slope 2 &
*y*-intercept 11. - Find the equation of the line with slope &
*y*-intercept -5.

**Answer Key**

- 3x â€“ 4y â€“ 7 = 0.
*Y = Â*x*Â*â€“- y = 4x
- 7x + y â€“ 12 = 0.
- y = 2x + 11
- y = x â€“ 5 or 3
*x*â€“ 7y â€“ 35=0.