Find the value of k so that the line through the given points has slope m.

(2k, 3), (1, k); m = 2

slope = (y2 - y1)/(x2 - x1)

so, [(k-3)/(1-2k)] = 2

(k-3) = 2(1-2k)

=> k = 1

To find the value of k such that the line passing through the points (2k, 3) and (1, k) has a slope of m = 2, we can use the formula for calculating the slope of a line:

m = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) and (x₂, y₂) are the coordinates of the two given points.

Let's substitute the given points into the formula:

m = (k - 3) / (1 - 2k)

Now, we need to solve this equation for k.

Multiply both sides of the equation by (1 - 2k) to eliminate the denominator:

m(1 - 2k) = k - 3

Expand the left side:

m - 2mk = k - 3

Rearrange the equation:

k - 2mk = m - 3

Factor out k:

k(1 - 2m) = m - 3

Divide both sides of the equation by (1 - 2m):

k = (m - 3) / (1 - 2m)

Now, substitute the given slope m = 2 into the equation:

k = (2 - 3) / (1 - 2(2))

Simplify the expression:

k = (-1) / (1 - 4)

k = (-1) / (-3)

k = 1/3

Therefore, the value of k that satisfies the given conditions is 1/3.