A line passes through #(3 ,6 )# and #(4 ,8 )#. 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    ,6   )# and #(4    ,8   )#.  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?

James Atkins · Accepted Answer

#(1,-3)# #"one way is to establish the equation of the line passing"# #"through "(7,9)" and find points that lie on it"# #• " parallel lines have equal slopes"# #"find the slope (m) using the "color(blue)"gradient formula"# #color(red)(bar(ul(|color(white)(2/2)color(black)(m=(y_2-y_1)/(x_2-x_1))color(white)(2/2)|)))# #"let "(x_1,y_1)=(3,6)" and "(x_2,y_2)=(4,8)# #rArrm=(8-6)/(4-3)=2# #"the equation of the line through "(7,9)# #y=2x+blarrcolor(blue)"slope-intercept form"# #"to find b substitute "(7,9)" into the equation"# #9=14+brArrb=-5# #rArry=2x-5larrcolor(blue)"equation of line"# #"choosing any value for x and evaluate for y"# #x=1toy=2-5=-3# #rArr(1,-3)" is a point on the second line"#

Wyatt Anderson · Answer

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

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

Using the points (3, 6) and (4, 8) for the first line, the slope is:

$$ m = \frac{8 - 6}{4 - 3} = \frac{2}{1} = 2 $$

Since the second line is parallel to the first line, it must also have a slope of 2.

Now, using the point-slope form of the equation of a line $ y - y_1 = m(x - x_1) $, we can find another point that the second line may pass through.

Using the point (7, 9) and the slope $ m = 2 $, we have:

$$ y - 9 = 2(x - 7) $$

$$ y - 9 = 2x - 14 $$

$$ y = 2x - 14 + 9 $$

$$ y = 2x - 5 $$

So, another point that the second line may pass through if it is parallel to the first line is (0, -5).