A line passes through #(2 ,3 )# and #( 4, 1 )#. A second line passes through #( 7, 4 )#. 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  ,3 )# and #(  4,  1  )#.  A second line passes through #(  7,  4  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Theodore Abad · Accepted Answer

One other point D is (1,10)

Given : #A(2,30, B(4,1), C(7,4)# To find one point say D,

Slope of AB = Slope of CD = (say) m

m = (1-3) / (4-2) = -1#

Equation of CD is

#y - 4 = -1 * (x - 7)#

#y + x = 11#

When x =1, y = 11 - 1 = 10#

One other point D is (1,10)

Aaliyah Anderson · 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, we use the formula:

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

Substituting the given points $(2, 3)$ and $(4, 1)$ into the formula:

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

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

Now, using the point-slope form of the equation of a line:

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

We can use the point $(7, 4)$ and the slope $m = -1$ to find the equation of the second line:

$$ y - 4 = -1(x - 7) $$

$$ y - 4 = -x + 7 $$

$$ y = -x + 11 $$

This is the equation of the second line. To find another point on this line, we can choose any value for $x$ and solve for $y$. For example, let's choose $x = 10$:

$$ y = -10 + 11 $$

$$ y = 1 $$

So, when $x = 10$, $y = 1$. Therefore, another point that the second line may pass through if it is parallel to the first line is $(10, 1)$.