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

Aaliyah Anderson · Accepted Answer

One possible point would be #(6,-2)#

A line through #(2,8)# and #(3,5)# has a slope of #color(white)("XXX")(Deltay)/(Deltax)=(5-8)/(3-2)=(-3)/1#

Any line parallel to this will have the same slope.

Any point on a line through #(4,4)# with the form #color(white)("XXX")(4+k * Deltax,color(white)("XX")4+k * Deltay)# with a non-zero constant #k# and #Deltax=1, Deltay=-3# will have a slope of #color(white)("XXX")(k * Deltay)/(k * Deltax)=(k * (-3))/(k *1)=(-3)/1# and will therefore be parallel to the line through #(2,8)# and #(3,5)#

Arbitrarily picking #k=2#, one such point would be #color(white)("XXX")(4+2 * Deltax,color(white)("XX")4+2 * Deltay)# #color(white)("XXXXXX") = (4+2 * 1,color(white)("X")4+2 * (-3))# #color(white)("XXXXXX")=(6, -2)#

Evelyn Arboleda · Answer

undefined To find a point that the second line may pass through if it is parallel to the first line, we first calculate the slope of the first line using the given points (2, 8) and (3, 5). The formula for slope (m) between two points (x₁, y₁) and (x₂, y₂) is: m = (y₂ - y₁) / (x₂ - x₁).

Using the points (2, 8) and (3, 5):
m₁ = (5 - 8) / (3 - 2)
m₁ = -3 / 1
m₁ = -3

Since the second line is parallel to the first line, it will have the same slope. Therefore, the slope of the second line is also -3.

Now, using the slope-intercept form of a line (y = mx + b), where m is the slope and b is the y-intercept, we can find the equation of the second line passing through the point (4, 4).

Using the slope (-3) and the point (4, 4):
4 = -3(4) + b
4 = -12 + b
b = 16

So, the equation of the second line is y = -3x + 16. To find another point on this line, we can choose any x-value and substitute it into the equation to solve for the corresponding y-value.

For example, let's choose x = 6:
y = -3(6) + 16
y = -18 + 16
y = -2

Therefore, another point that the second line may pass through, if it is parallel to the first line, is (6, -2).