A line passes through #(1 ,5 )# and #(8 ,7 )#. A second line passes through #(3 ,6 )#. 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 #(1    ,5  )# and #(8   ,7   )#.  A second line passes through #(3   ,6  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Maverick Smith · Accepted Answer

#(10,8)#

I didn't use math to figure this out, but I'll explain both ways. First the easy way.

To get from #(1,5)# to #(3,6)#, you increase #x# by #2# and #y# by #1#, so if you do the same thing to #(8,7)#, you get #(10,8)#. If you move both points by the same amount, the slope stays the same.

Mathematically, you would have to find the slope of the line made by #(1,5)# and #(8,7)#. So you do the slope formula, and you get

#"slope" = m = (7-5)/(8-1#

which equals #2/9#. Then you use the point-slope formula and input #(3,6)# for the #x# and #y# variables and #2/9# for the #m# variable.

You end up with

#y-6=(2/7)(x-3)#

which, when you solve for #y#, becomes

#y=2/7x+36/7#

(I kept the fractions so it would be easier to solve).

Then just pick an #x# value and plug it in yo get your #y# and you're done. Yay! (You'll notice that plugging in #10# gets you #56/7#, which is #8#)

Savannah Aguilar · Answer

undefined To find another point that the second line may pass through if it is parallel to the first line, we use the concept that parallel lines have the same slope.

First, we find the slope of the first line passing through points (1, 5) and (8, 7). The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

Using the given points:
m = (7 - 5) / (8 - 1)
m = 2 / 7

Now, since the second line is parallel to the first line, it must have the same slope. Thus, the second line passing through the point (3, 6) and having the same slope of 2/7 can be found by substituting the slope and one point into the point-slope form of a line.

The point-slope form of a line is:
y - y1 = m(x - x1)

Substituting the values:
y - 6 = (2/7)(x - 3)

Now, we can solve for y to find the equation of the second line. We can also choose any value for x to find the corresponding y-coordinate. For simplicity, let's choose x = 0:

y - 6 = (2/7)(0 - 3)
y - 6 = (2/7)(-3)
y - 6 = -6/7
y = -6/7 + 6
y ≈ 41/7

So, one other point that the second line may pass through, given that it is parallel to the first line, is approximately (0, 41/7) or (0, 5.857).