Various equation forms, continued.


We now will provide proof that the line shown in the graph can be represented by an equation in three different forms. Since all the equation forms are equal to one another they can be converted from one form to another and still represent the line. Notice also that if we are given two sets of (x,y) points on the line we can calculate the line slope and y intercept. This is all the information we need to fully describe the line in any of the three equation formats.

         Convert two points (3,2) and (8,4) to point slope equation form.

(y2 - y1) / (x 2 - x1)          :: Find the slope of the line between two points.
(4 - 2) / (8 - 3)                :: Enter the values of the two points.
2/5               :: This is the slope or m of the line.
(y
- 2) / (x - 3) = 2/5   :: Replace the y2 = 4 and x 2 = 8 points with y and x.
((y
-2)/(x -3)) * (x -3) = 2/5 * (x -3) :: Multiply both sides by (x-3)
y - 2 = 2/5x + .8    :: Point slope form
y - y1 = m(x - x1)

         Convert the point slope form of the equation to the slope intercept form.

y - 2 = 2/5(x - 3)              :: Point slope form y - y1 = m(x - x1)
y - 2 +2 = 2/5(x - 3) +2    :: Add +2 to each side
y = 2/5*x - 2/5*3 + 2   :: Multiply the term (x-3) by 2/5
y = 2/5*x - 1.2 + 2           :: Now add -1.2 and +2
y = 2/5x - .8              :: Change the decimal to a fraction
y = 2/5x - 4/5             :: Slope intercept form
y = mx + b

         Convert the slope intercept form to the standard form.

y = 2/5x - 4/5                     :: Slope intercept form y = mx + b
y * 5 = (2/5x)*5 - (4/5)*5     :: Multiply each side by 5
5y = 2x - 4      :: Move 2x to the left side and change the sign.
5y - 2x = - 4                  :: Standard form
Ax + By = C

         Convert standard form to point slope form.

2x - 5y = - 4                  :: Standard form of Ax + By = C
2x + 4 = 5y :: Move the -4 to the left and 5y to the right.
5y = 2x + 4 :: Reverse the order to get 5y on the left.
5y/5 = (2x + 4)/5            :: Convert the fraction 4/5 to a decimal
y = 2/5x + .8 :: Point slope form.
y - y1 = m(x - x1)


         Given: (x1 , y1) and (x2, y2) representing two points on a line;
use
(y - y1)/(x - x1) = (y2 - y1) / (x2 - x1) to solve for the lines equation in standard form.

         Given:  (x, y) and m representing one point on the line and the slope of the line;
use
(y - y1)/(x - x1) = m and solve for y giving the slope intercept equation form.

       Given: m and b representing the slope of the line and the y intercept;
use
y = mx + b to give the slope intercept form of the equation.


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