A line passes through #(6 ,2 )# and #(5 ,3 )#. A second line passes through #(7 ,7 )#. 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 #(6    ,2   )# and #(5   ,3   )#.  A second line passes through #(7    ,7  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Allison Allen · Accepted Answer

The direction vector for the first line is #(6,2)-(5,3)=(1,-1)# so the second, parallel line is #(7,7)+t(1,-1)# so includes #(8,6), (9,5), (6,8), etc.#

Carson Arnold · Answer

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

The slope of the first line passing through (6,2) and (5,3) can be found using the formula:

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

Substituting the coordinates of the points into the formula:

$$ m = \frac{{3 - 2}}{{5 - 6}} $$
$$ m = \frac{{1}}{{-1}} $$
$$ m = -1 $$

So, the slope of the first line is -1.

Since the second line is parallel to the first line, it must also have a slope of -1.

Using the point-slope form of a line, we can find the equation of the second line passing through (7,7) with a slope of -1:

$$ y - y_1 = m(x - x_1) $$

Substituting $ x_1 = 7 $, $ y_1 = 7 $, and $ m = -1 $:

$$ y - 7 = -1(x - 7) $$
$$ y - 7 = -x + 7 $$
$$ y = -x + 14 $$

So, the second line, if parallel to the first, passes through any point that satisfies the equation $ y = -x + 14 $. One such point is (8,6).