A line passes through #(4 ,3 )# and #(2 ,5 )#. A second line passes through #(5 ,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 #(4    ,3  )# and #(2   ,5   )#.  A second line passes through #(5    ,6  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Theodore Abad · Accepted Answer

(3,8)

So we first have to find the direction vector between (2,5) and (4,3) #(2,5)-(4,3)=(-2,2)# We know that a vector equation is made up of a position vector and a direction vector. We know that (5,6) is a position on the vector equation so we can use that as our position vector and we know that it is parallel the other line so we can use that direction vector #(x,y)=(5,6)+s(-2,2)# To find another point on the line just substitute any number into s apart from 0 so lets choose 1 #(x,y)=(5,6)+1(-2,2)=(3,8)# So (3,8) is another another point.

Maverick Smith · Answer

undefined If the second line is parallel to the first line, then it will have the same slope as the first line. To find the slope of the first line, you can use the formula:

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

Using the points (4, 3) and (2, 5) for the first line:

$$ m = \frac{5 - 3}{2 - 4} = \frac{2}{-2} = -1 $$

Now that you know the slope of the first line is -1, you can use this slope and the given point (5, 6) for the second line to find the equation of the second line in point-slope form:

$$ y - y_1 = m(x - x_1) $$
$$ y - 6 = -1(x - 5) $$
$$ y - 6 = -x + 5 $$
$$ y = -x + 11 $$

This equation represents the second line. To find another point on this line, you can choose any value for $ x $ and calculate the corresponding $ y $ value using the equation. For example, if $ x = 6 $:

$$ y = -6 + 11 = 5 $$

So, when $ x = 6 $, $ y = 5 $. Therefore, another point through which the second line may pass is (6, 5).