A line passes through #(2 ,5 )# and #(0 ,1 )#. 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 #(2    ,5  )# and #(0   ,1   )#.  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?

Sadie Allen · Accepted Answer

#p(8,3)# for example

First, find the slope of the first line. #m=(y_1-y_2)/(x_1-x_2)=(5-1)/(2-0)=4/2=2# Because line 1 is parallel to line 2, they have the same slope. Knowing that the slope of line 2 is #m=2=color(red)(2)/color(blue)(1)# as well, another point would be #p(7+color(blue)(1),1+color(red)(2))=p(8,3)#

Ella Armstrong · Answer

undefined If the second line is parallel to the first line, it means their slopes are equal. To find the slope of the first line passing through the points (2, 5) and (0, 1), we use the formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

$$ m = \frac{1 - 5}{0 - 2} = \frac{-4}{-2} = 2 $$

Now, as the second line is parallel to the first, its slope should also be 2. Using the point-slope form of a line, we can write the equation of the second line passing through the point (7, 1) as:

$$ y - 1 = 2(x - 7) $$

Expanding this equation, we get:

$$ y - 1 = 2x - 14 $$

$$ y = 2x - 13 $$

To find another point on this line, we can pick any value of x and substitute it into the equation to solve for y. Let's choose x = 0:

$$ y = 2(0) - 13 $$

$$ y = -13 $$

So, another point that the second line may pass through is (0, -13).