A line passes through #(6 ,4 )# and #(9 ,0 )#. A second line passes through #(3 ,4 )#. 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 #(6    ,4  )# and #(9   ,0   )#.  A second line passes through #(3   ,4  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Jayden Jackson · Accepted Answer

#color(green)("One other point on the second line is " (6, 0)#

#"Slope of line 1 " = m = (y_2 - y_1) / (x_2 - x_1) = (0-4) / (9-6) = -4/3#

As #2^(nd)# line is parallel to the first line, slope of second line also #-4/3#

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

#(y - 4) = -4/3 * (x - 3)#

#3y - 12 = -4x + 12#

#4x + 3y = 24#

Let y = 0. Then x = 6#

#color(green)("One other point on the second line is " (6, 0)#

Wyatt Anderson · Answer

undefined If the second line is parallel to the first line passing through the points (6, 4) and (9, 0), it must have the same slope as the first line.

1. Calculate the slope of the first line using the given points:
$$ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{0 - 4}}{{9 - 6}} = \frac{{-4}}{{3}} $$

2. Since the second line is parallel, it must also have a slope of $ -\frac{4}{3} $.

3. Using the point-slope form of the line, we can find another point on the second line:
$$ y - y_1 = m(x - x_1) $$
$$ y - 4 = -\frac{4}{3}(x - 3) $$

4. Choose a value for $x$ (other than 3) and solve for $y$:
$$ y - 4 = -\frac{4}{3}(x - 3) $$
$$ y - 4 = -\frac{4}{3}x + 4 $$
$$ y = -\frac{4}{3}x + 8 $$

So, if $x = 0$, then:
$$ y = -\frac{4}{3}(0) + 8 = 8 $$

Thus, another point through which the second line passes, parallel to the first line, is (0, 8).