A line passes through #(1 ,5 )# and #(8 ,3 )#. A second line passes through #(7 ,6 )#. 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 #(1    ,5  )# and #(8   ,3   )#.  A second line passes through #(7   ,6  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Allison Allen · Accepted Answer

#color(green)("One other point on the second line is " (21,2)#

#"Slope of line 1 " = m = (y_2 - y_1) / (x_2 - x_1) = (3-5) / (8-1) = -2/7#

As #2^(nd)# line is parallel to the first line, slope of second line also #-2/7#

#"Equation of second line in point - slope form is "

#(y - 6) = -2/7 * (x - 7)#

#7y - 42 = -2x + 14#

#2x + 7y = 56#

Let y = 2. Then x = 21#

#color(green)("One other point on the second line is " (21,2)#

Aaliyah Anderson · Answer

undefined If the second line is parallel to the first line, it means they have the same slope. To find a point through which the second line passes, we can use the slope of the first line and the given point on the second line.

The slope of the first line can be found using the coordinates of the given points (1, 5) and (8, 3).

$$ \text{Slope} = \frac{{\text{change in } y}}{{\text{change in } x}} = \frac{{3 - 5}}{{8 - 1}} = \frac{{-2}}{{7}} $$

Since the second line is parallel to the first, it also has a slope of $ -2/7 $. Now, using the given point (7, 6) on the second line and the slope, we can find the other point:

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

$$ y - 6 = \frac{{-2}}{{7}}(x - 7) $$

$$ y - 6 = -\frac{{2}}{{7}}x + 2 $$

$$ y = -\frac{{2}}{{7}}x + 8 $$

So, the second line passes through the point $(x, y) = (0, 8)$.