A line passes through #(2 ,3 )# and #( 4, 5 )#. 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?

1 Answer
Mar 14, 2018

See a solution process below:

Explanation:

First, we need to determine the slope of the first line. The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.

Substituting the values from the points in the problem gives:

#m = (color(red)(5) - color(blue)(3))/(color(red)(4) - color(blue)(2)) = 2/2 = 1#

Because the problem states the two lines are parallel therefore the second line will have the same slope of #m = 1#

Because we have a slope and one point for the second line we can write an equation for the line using the point-slope formula. The point-slope form of a linear equation is: #(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))#

Where #(color(blue)(x_1), color(blue)(y_1))# is a point on the line and #color(red)(m)# is the slope.

Substituting the values from the point in the problem and slope we calculated gives:

#(y - color(blue)(4)) = color(red)(1)(x - color(blue)(7))#

#y - color(blue)(4) = x - color(blue)(7)#

#y - color(blue)(4) + 4 = x - color(blue)(7) + 4#

#y - 0 = x - 3#

#y = x - 3#

To find another point on this second line substitute any number you want for #x# (other than 7) and calculate #y#.

I am going to choose an easy number - #0#

#y = x - 3# becomes:

#y = 0 - 3#

#y = -3#

So the point would be: #(0, -3)#