A line passes through #(3 ,2 )# and #(7 ,3 )#. A second line passes through #(8 , 1 )#. What is one other point that the second line may pass through if it is parallel to the first line?

Lines are parallel if they have the same slope.

So, to answer this question, we need to:
1. Find the slope of the first line
2. Apply that slope to the second line to find a corresponding point

SOLUTION

1. Slope of first line
   Slope or #m# can be found by: #m=(Delta y)/(Delta x)=(y_2 - y_1)/(x_2 - x_1)#
   Using the given information, #m=(3 - 2)/(7 - 3)=color(blue)(1/4)#
   Both lines will have a slope of #1/4#
2. Using #m=1/4#, find a point on the second line

Here, there are infinite possibilities for points on the second line.
If we use the point #(8,1)# as #(x_1, y_1)#, then #(x_2, y_2)# must
satisfy the equation

#m=1/4=(y_2 - y_1)/(x_2 - x_1)#

The easiest solutions can be found by using #Delta y=1# and #Delta x=4#

#Delta y = 1#
#y_2 - y_1 = 1#
#y_2 - 1 = 1#
#color(blue)(y_2 = 2)#

#Delta x = 4#
#x_2 - x_1 = 4#
#x_2 - 8 = 4#
#color(blue)(x_2 = 12)#

One solution is #(12, 2)#

Another solution can be found by solving
#1 - y = 1# and #8 - x = 4#

But as stated earlier, there are many, many more valid solutions.