Question

A(2,8), B(6,4) and C(-6,y) are collinear points find y?

Answer 1

`y=16`

Explanation:

If a set of points are collinear the belong to the same straight line, whose generale equation is `y=mx+q`
If we apply the equation to the point A we have:
`8=2m+q`
If we apply the equation to the point B we have:
`4=6m+q`
If we put this two equation in a system we can find the equation of the straight line:

1. Find `m` in the first eq.
   `m=(8-q)/2`
2. Replace `m` in the second eq. and find `q`
   `4=6(8-q)/2=>4=3(8-q)+q=>4=24-3q+q=>-20=-2q=>q=10`
3. Replace `q` in the first eq.
   `m=(8-10)/2=-1`
   Now we have the equation of the straight line:
   `y=-x+10`
   If we replace C coordinates in the equation we have:
   `y=6+10=>y=16`

Answer 2

`16`.

Explanation:

Prerequisite :

`"The points "(x_1,y_1),(x_2,y_2) and (x_3,y_3)" are collinear"`

`hArr |(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|=0`.

Therefore, in our Problem, `|(2,8,1),(6,4,1),(-6,y,1)|=0`,

`rArr 2(4-y)-8{6-(-6)}+1{6y-(-24)}=0`,

`rArr 8-2y-96+6y+24=0`,

`rArr 4y=64`,

`rArr y=16,` as Respected Lorenzo D. has already derived!.

Answer 3

`P_C->(x_c,y_c)=(-6,+16)`

Full details shown. With practice you will be able to do this calculation type with very few lines.

Explanation:

`color(blue)("The meaning of 'collinear'")`

Lets split it into two parts

`color(brown)("co"->"together".` Think about the word cooperate
`color(white)("ddddddddddddd")`So this is 'together and operate.'
`color(white)("ddddddddddddd")`So you are doing some operation (activity)
`color(white)("ddddddddddddd")`together

`color(brown)("liniear".->color(white)("d")` In a strait line.

`color(brown)("collinear")->` co =together, linear =on a strait line.

`color(brown)("So all the points are on a strait line")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

`color(purple)("Determine the gradient (slope)")`

The gradient for part is the same as the gradient for all of it

Gradient (slope) `->("change in y")/("change in x")`

Set point `P_A->(x_a,y_a)=(2,8)`
Set point `P_B->(x_b,y_b)=(6,4)`
Set point `P_C->(x_c,y_c)=(-6,y_c)`

The gradient ALWAYS reads left to right on the x-axis (for standard form)

So we read from `P_A " to " P_B` thus the we have:

Set gradient`-> m="last "-" first"`

`color(white)("d")"gradient " -> m=color(white)("d")P_Bcolor(white)("d")-color(white)("d")P_A`

`color(white)("dddddddddddd")m=color(white)("d,")(y_b-y_a)/(x_b-x_a)`

`color(white)(dddddddddddddddddddd") (4-8)/(6-2) = -4/4=-1`

Negative 1 means that the slope (gradient) is downward as you read left to right. For 1 across there is 1 down.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(purple)("Determine the value of "y)`

Determined that `m=-1` so by direct comparison

`P_C-P_A =m = (y_c-y_a)/ (x_c-x_a) = -1`

`color(white)("dddddddddddd.d") (y_c-8)/ (-6-2) = -1`

`color(white)("dddddddddddddd.") (y_c-8)/ (-8) = -1`

Multiply both sides by (-8)

`color(white)("ddddddddddddddd.") y_c-8 = +8`

Add 8 to both sides

`color(white)("ddddddddddddddddd.")y_c color(white)("d")=+16`