Question

If the points (2, 4), (5, k), and (8, 20) are on the same line, what is the value of k?

Answer 1

The value of k, ensuring that the points (2, 4), (5, k), and (8, 20) lie on the same line, is calculated using the slope equation and is found to be 12.

Step-by-step explanation:

To find the value of k for the points to lie on the same line, we need to use the concept of slope. The slope of a line passing through two points, (x1, y1) and (x2, y2), is calculated as (y2 - y1) / (x2 - x1). Since the points (2, 4), (5, k), and (8, 20) should all lie on the same line, they should have the same slope.

So, let's calculate the slope of the line through (2, 4) and (8, 20):

Slope = (20 - 4) / (8 - 2) = 16 / 6 = 8 / 3

Now we need to ensure that the slope between (2, 4) and (5, k) is the same:

Slope = (k - 4) / (5 - 2) = (k - 4) / 3

Setting the slopes equal to each other gives us:

(k - 4) / 3 = 8 / 3

By multiplying both sides by 3 and then adding 4 to both sides, we find:

k = 8 + 4 = 12

Therefore, the value of k is 12.

Answer 2

k = 12

Step-by-step explanation:

Calculate the slope m using the slope formula

m =
`(y_(2)-y_(1) )/(x_(2)-x_(1) )`

with (x₁, y₁ ) = (2, 4) and (x₂, y₂ ) = (8, 20)

m =
`(20-4)/(8-2)` =
`(16)/(6)` =
`(8)/(3)`

Repeat using (5, k) as one of the points and equate to
`(8)/(3)`

(x₁, y₁ ) = (2, 4) and (x₂, y₂ ) = (5, k) , then

m =
`(k-4)/(5-2)` =
`(k-4)/(3)` =
`(8)/(3)` ( cross- multiply )

3(k - 4) = 24 ( divide both sides by 3 )

k - 4 = 8 ( add 4 to both sides )

k = 12