A line passes through #(4 ,3 )# and #(2 ,9 )#. A second line passes through #(7 ,1 )#. What is one other point that the second line may pass through if it is parallel to the first line?

1 Answer
Math Tutor
Jun 6, 2018

From the slope and the given point, we can determine many points that will make this line parallel, including: #(6,4)#

Explanation:

For the lines to be parallel, it implies that they have the same slope. So, let's first determine what is that slope.
For the first line we have 2 points, so we can determine the slope of that line as:
# m = (Delta y)/(Delta x) = (y_2-y_1)/(x_2-x_1) = (9-3)/(2-4) = 6/-2 = -3#

The slope of the other line must be the same: #m=-3#
As stated in the question, the line goes through point #(7,1)#. Let's call that point #P_1#.
For our second point, #P_2#, we can choose any value for #x_2#, but we need to determine the corresponding value for #y_2#.
So, if we choose, say, #x_2=6#, what is the value of #y_2# ?
We can apply the same equation for slope, and our only unknown is #y_2#:
# m = (Delta y)/(Delta x) = (y_2-y_1)/(x_2-x_1)#
#-3= (y_2 - 1)/(6-7)#
#-3= (y_2-1)/-1#
#3 = y_2-1#,
#y_2 = 4#
So, a valid point in that parallel line is #(6,4)#

There is an infinite number of possible solutions, as long as we maintain #m=-3#. Another example would be #(8,-2)#

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