2.2 Lines, 2.3 Lines, Finding the Equation of a Line

Finding the Equation of a Line given two points on the line

Example:  Find the equation of a line in slope intercept form given the line passes through the two points (5,-3) and (6,-1).

Solution:

First find the slope of the line.

Choose one of the points to be   ( x_1, y_1) and choose the other point to be   ( x_2, y_2).

I will choose   ( 5, -3)  to be   ( x_1, y_1)  and choose   ( 6, -1) to be   ( x_2, y_2).

Substitute these values into the slope formula and simplify.

  m= {y_2-y_1} / {x_2-x_1} ={-1-(-3)}/{6-5}={-1+3}/{1} =2/1=2

The slope of the line containing the points   ( 5, -3) and   ( 6, -1)  is m= 2.

Then, use the point-slope formula of the line to start building the line.  m represents the slope of the line and you can use (x_1,y_1) or (x_2,y_2) as the point on the line.

Point-slope formula: y-y_1 = m(x-x_1)

m=2 and  (5,-3)

Substitute the values into the formula.

 y-(-3) = 2(x-5)

Since the instructions ask to write the equation in slope intercept form (y=mx+b) we will simplify and write the equation with y by itself on one side.

 y-(-3) = 2(x-5)

 y+3 = 2x-10

 y+3-3 = 5x-10-3

 y = 5x-13

The equation of a line in slope intercept form passing through the two points (5,-3) and (6,-1) is  y = 5x-13.