Md.Nazrul Islam wrote:In a co-ordinate system , if three points (5,3)(x,y) and (3,2)lie on a same line ,fine the value of X.

Since the three points lie on the same line, so the slope of any two points will be the same.
Now slope of a line passing through (x1, y1) and (x2, y2) = (y2 - y1)/(x2 - x1)

Slope of line passing through (5, 3) and (x , y) = (y - 3)/(x - 5)
Slope of line passing through (x, y) and (3 , 2) = (2 - y)/(3 - x)
Now, Slope of line passing through (5, 3) and (x , y) = Slope of line passing through (x, y) and (3 , 2)
(y - 3)/(x - 5) = (2 - y)/(3 - x)
(y - 3)(3 - x) = (2 - y)(x - 5)
3y - 9 - xy + 3x = 2x - xy - 10 + 5y
x - 2y = -1 ... Equation 1

Similarly, slope of line through (5, 3) and (3, 2) = Slope of line passing through (x, y) and (3 , 2)
(2 - 3)/(3 - 5) = (2 - y)/(3 - x)
(-1/-2) = (2 - y)/(3 - x)
1/2 = (2 - y)/(3 - x)
3 - x = 2(2 - y)
3 - x = 4 - 2y
x - 2y = -1 ... Equation 2

Both equations 1 and 2, give the same equation, so we can have infinite solutions to this, and hence infinite values for x.