A line passes through #(3 ,4 )# and #(4 ,7 )#. A second line passes through #(7 ,9 )#. 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 #(3    ,4   )# and #(4    ,7   )#.  A second line passes through #(7    ,9  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Ella Armstrong · Accepted Answer

Any point on the line #y = 3x-12# is a viable answer.

Slope formula from two points: #m = (y_2-y_1)/(x_2-x_1)# Slope of the given line: #m = (7-4)/(4-3)=3/1=3# The equation of this line using point-slope form: #y-y_1=m(x-x_1)=># #y-7=3(x-4)# #y-7=3x-12# #y=3x-5# Since the unknown line is parallel to this one, it has the same slope. Using the newfound slope, we can craft an equation for the new line: #y-y_1=m(x-x_1)=># #y-9=3(x-7)# #y-9=3x-21# #y=3x-12# This is the "second line" that the question mentions. Now, we just need to pick a point anywhere on this line by plugging in an #x#-value and get a #y#-value back. I'll pick 6 for an #x#-value: #y=3x-12=># #y=3(6)-12# #y=18-12# #y=6# A point on the line is #(6, 6)#.

Noah Atkinson · Answer

undefined Since 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 (3, 4) and (4, 7) is given by:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 4}{4 - 3} = \frac{3}{1} = 3 $$

Therefore, the second line passing through (7, 9) and parallel to the first line will also have a slope of 3.

Using the point-slope form of a linear equation, the equation of the second line passing through (7, 9) with slope 3 is:

$$ y - 9 = 3(x - 7) $$

Expanding this equation gives:

$$ y - 9 = 3x - 21 $$

$$ y = 3x - 12 $$

To find another point on this line, we can choose any value of $ x $ and calculate the corresponding $ y $. Let's choose $ x = 0 $:

$$ y = 3(0) - 12 $$

$$ y = -12 $$

Therefore, another point that the second line may pass through is (0, -12).