A line passes through #(9 ,3 )# and #( 3, 5 )#. A second line passes through #( 7, 8 )#. 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 #(9  ,3 )# and #(  3,  5  )#.  A second line passes through #(  7,  8  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Ella Armstrong · Accepted Answer

#(0, 17)#

Parallel lines have the same slope.

Get the slope using the points passed through by the first line

#m = (y_1 - y_2)/(x_1 - x_2)#

#P_1: (9, 3)# #P_2: (3, 5)#

#=> m = (3 - 5)/(9 - 3)#

#=> m = -2/6 = -1/3#

Get the equation of the second line

#y = mx + b#

#y = -1/3x + b#

Substitute the point passed through by the second line to get the y-intercept

#P_1': (7, 8)#

#8 = -1/3(7) + b#

#24 = -7 + b#

#b = 17#

Hence, the equation of the second line is

#y = -1/3x + 17#

To determine another point that the line passes through, simply select a desired value for either #x# or #y#. Substitute the desired value into the equation to get its corresponding #y# (or #x#).

For simplicity, let's use #x = 0#.

#y = -1/3(0) + 17#

#y = 17#

Hence, we have

#P_2': (0, 17)#

James Atkins · Answer

undefined If the second line is parallel to the first line, it will have the same slope as the first line.

The slope of the first line passing through (9, 3) and (3, 5) is:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{3 - 9} = \frac{2}{-6} = -\frac{1}{3} $$

Since the second line is parallel, it will also have a slope of -1/3. Using the point-slope form of a linear equation $(y - y_1) = m(x - x_1)$, where $m = -\frac{1}{3}$ and $(x_1, y_1) = (7, 8)$, we can find another point on the second line:

$$ (y - 8) = -\frac{1}{3}(x - 7) $$

$$ y - 8 = -\frac{1}{3}x + \frac{7}{3} $$

$$ y = -\frac{1}{3}x + \frac{7}{3} + 8 $$

$$ y = -\frac{1}{3}x + \frac{31}{3} $$

Therefore, another point on the second line is (0, 31/3).