SOLUTION: A pencil of lines described by y=mx+3m-2

Let m=a

y  = a + 3a - 2

let m=b where a and b are not equal.

y = bx + 3b - 2

Solve the system:



ax + 3a - 2 = bx + 3b - 2

ax - bx = -3a + 3b

x(a - b) = -3(a - b)

Since a is not equal to b we may divide through by (a - b) and
get:

x = -3

Substituting in

y = ax + 3a - 2
y = a(-3) + 3a - 2
y = -3a + 3a - 2
y = -2

So we have a pencil of lines all going through the point (x,y) = (-3,-2).



[However the pencil of lines does not include the vertical line 
through (-3,-2) because there is no way to get the equation x=-3 
from the equation y = mx + 3m - 2, by substituting a value for m] 

Edwin