# If l is a line through the points (2,5) and (4,6) what is the value of k so that the point of coordinates (7,k) is on l.

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W have the points: (2,5) and (4,6) passes through a lint L .

We need to determine k values such that ( 7, k) is on the line:

First we will determine the equation for the line L:

( y- y1) = m (x-x1) where (x1,y1) is any point of the line and m is the slope:

==> ( y-5) = m (x-2) ............(1)

Now we will determine the slope:

m= ( y2-y1)/(x2-x1) = ( 6-5)(4-2) = 1/2

==> m = 1/2

==> (y-5) = (1/2) ( x-2)

==> y-5 = (1/2)x - 1

==> y= (1/2)x - 1 +5

==> y= (1/2)x + 4

Now we know that the point ( 7, k ) passes through the line:

Then ( 7,k) should verify the equation for the line:

==> k= (1/2)*7 + 4

==? k = 7/2 + 4 = 15/2 = 7.5

**==> k = 7.5**

We have to find the equation of the line through (2,5) and (4,6).

Now we know that the line through (x1 , y1) and (x2, y2) is given by y - y1 = [(y2 - y1)/ (x2 - x1)] ( x- x1)

=> y - 5 = [( 6 - 5) / ( 4 - 2)](x -2)

=> y - 5 = (1 / 2) ( x-2)

If the point (7 , k) lies on this line we can determine k by substituting 7 and k in the equation of the line.

Hence k - 5 = (1 / 2) ( 7 - 2)

=> k - 5 = 5/2

=> k = 5/2 + 5

=> k = 15/2

**Therefore the required value of k is 15/2.**

The line through the points (x1,y1) and (x2,y2) is given by:

y-y1 = {(y2-y1)/(x2-x1)}(x-x1).

So the line through the given points (2,5) and 4,6) is given by:

y-5 = {(6-5)/(4-2)}(x-2)

y-4 = (1/2)(x-2)

y = (1/2)x -2+4.

y = (1/2)x +4. So the point (7,k) should satisfy this equation as it lies on this line by data.

So k = (1/2)7+2 = 5.5.

Therefore k = 5.5 so that it lies on the line y = (1/2)x +4.

We'll determine the equation of the line that passes through the points: (2,5) and (4,6)

(x2 - x1)/(x - x1) = (y2 - y1)/(y - y1)

We'll identify the cordinates:

x1 = 2, x2 = 4

y1 = 5, y2 = 6

We'll substitute into the formula:

(4-2)/(x - 2) = (6-5)/(y - 5)

2/(x-2) = 1/(y-5)

We'll cross multiply:

2y - 10 = x - 2

2y = x + 8

y = x/2 + 4

If the point (7,k) is on the line ,then it's coordinates verify the equation of the line:

k = 7/2 + 4

k = (7+8)/2

**k = 15/2**