A line passes through #(2 ,8 )# and #(4 ,5 )#. A second line passes through #(3 ,5 )#. 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 #(2    ,8   )# and #(4    ,5   )#.  A second line passes through #(3    ,5  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Sophia Alfred · Accepted Answer

#(0,9.5) and (19/3,0)#

Parallel lines have same slope/gradient.

Slope of first line = #m# = #[(y_2 - y_1)/(x_2 - x_1)]#

= #[(5-8)/(4-2)]# = #(-3/2)#

Therefore slope of another line is too #-(3/2)#.

Equation of another line :

#(y - y_1) = m (x - x1)#

#(y - 5) = -3/2 (x - x_1)#

#2 xx (y - 5) = -3 xx (x-3)#

#2y - 10 = -3x +9#

#2y + 3x = 19#

Now by trial and error method we can put two values of #x and y# such that these values satisfy the above equation. Best way is to put #x = 0# and # y =0#.

So the other points could be

(0,9.5) and (#19/3#, 0)

Carson Arnold · Answer

undefined If the second line is parallel to the first line passing through the points $(2, 8)$ and $(4, 5)$, then it must have the same slope as the first line.

The slope of the first line can be calculated using the formula:

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

For the first line passing through $(2, 8)$ and $(4, 5)$:

$$ m = \frac{5 - 8}{4 - 2} = \frac{-3}{2} $$

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

Now, we use the point-slope form of a line equation:

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

Using the point $(3, 5)$ and the slope $ -\frac{3}{2} $, we can find the equation of the second line:

$$ y - 5 = -\frac{3}{2}(x - 3) $$

$$ y - 5 = -\frac{3}{2}x + \frac{9}{2} $$

$$ y = -\frac{3}{2}x + \frac{9}{2} + 5 $$

$$ y = -\frac{3}{2}x + \frac{19}{2} $$

Now, to find another point on this line, we can choose any $x$ value and solve for the corresponding $y$ value. For simplicity, let's choose $x = 1$:

$$ y = -\frac{3}{2}(1) + \frac{19}{2} $$

$$ y = -\frac{3}{2} + \frac{19}{2} $$

$$ y = 8 $$

So, the other point that the second line may pass through, if it is parallel to the first line, is $(1, 8)$.