Expert: Paul Klarreich - 12/29/2011

Question
Line 1 has a slope of M, and y intercept of (0,2). Line 2 has a slope of 2 and a y intercept of (0,M). Find the coordinates of the point of intersection

Answer
Questioner:Flo
Country:Connecticut, United States
Category:Advanced Math
Private:Yes   <<<<<<<<<<<<<<< changed
Question:

Line 1 has a slope of M, and y intercept of (0,2). Line 2 has a slope of 2 and a y intercept of (0,M). Find the coordinates of the point of intersection
.............................................
L1 has equation:  y = mx + 2
L2 has equation:  y = 2x + m

Set:  mx + 2 = 2x + m
Solve for x:

2 + m = (2 - m)x
   2 + m
x = -----
   2 - m

Go back to the equations and substitute for x:
     2 + m
y = m(-----) + 2
     2 - m
.................
     2 + m
y = 2(-----) + m
     2 - m
.................
Set those equal to each other:

 2 + m          2 + m
2(-----) + m = m(-----) + 2
 2 - m          2 - m

Multiply through by (2-m) to clear fractions:
   
2(2 + m) + m(2 - m) = m(2 + m) + 2(2 - m)

4 + 2m + 2m - m^2 = 2m + m^2 + 4 - 2m

   2m + 2m - m^2 = 2m + m^2     - 2m

   2m + 2m - m^2 =      m^2

   4m  =  2m^2

   0 =  2m^2 - 4m

m = 0  or m = 2
........................
m = 0:
L1 has equation:  y = 2
L2 has equation:  y = 2x
Point is y = 2, x = 1;  (1,2)
........................
m = 2
L1 has equation:  y = 2x + 2
L2 has equation:  y = 2x + 2

These are the same line, so all points are intersections.