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?

Question

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?

Avery Allen · Accepted Answer

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

Sadie Allen · Answer

undefined To find a point through which the second line may pass if it is parallel to the first line, we first need to determine the slope of the first line. Using the formula for slope, which is (change in y) / (change in x), we calculate:

Slope of the first line = (9 - 3) / (2 - 4) = 6 / -2 = -3

Since the second line is parallel to the first line, it will have the same slope. Now, using the point-slope form of the equation of a line, y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can find the equation of the second line. Let's choose the point (7, 1) from the given point:

y - 1 = -3(x - 7)

Expanding and simplifying:

y - 1 = -3x + 21
y = -3x + 22

Now, to find another point on this line, we can arbitrarily choose a value for x and solve for y. Let's choose x = 5:

y = -3(5) + 22
y = -15 + 22
y = 7

So, the point (5, 7) lies on the second line, which is parallel to the first line passing through (4, 3) and (2, 9).