If the slope of a line joining P(8, k) and Q(1-5k, 3) is 1/7, then the midpoint of PQ is:

To find the midpoint of a line segment PQ with endpoints P(8, k) and Q(1-5k, 3), we can use the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

First, let's find the coordinates of point Q:

x-coordinate of Q = 1 - 5k
y-coordinate of Q = 3

Now, we know that the slope of the line joining P and Q is 1/7. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

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

Substitute the coordinates of P and Q into the slope formula:

1/7 = (3 - k) / (1 - 5k - 8)
1/7 = (3 - k) / (-7 - 5k)
1/7 = (3 - k) / (-7 - 5k)
7(3 - k) = -7 - 5k
21 - 7k = -7 - 5k
21 + 7 = -7 + 5k
28 = -7 + 5k
28 + 7 = 5k
35 = 5k
k = 7

Now that we have found the value of k, we can find the coordinates of point Q:

x-coordinate of Q = 1 - 5(7) = 1 - 35 = -34
y-coordinate of Q = 3

Now, we can find the midpoint of PQ:

Midpoint = ((8 + (-34))/2, (7 + 3)/2)
Midpoint = (-26/2, 10/2)
Midpoint = (-13, 5)

Therefore, the midpoint of PQ is (-13, 5).

The slope of a line joining two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).

Using this formula, we have:

Slope of PQ = (3 - k) / ((1-5k) - 8) = 1/7

Simplifying the denominator:

Slope of PQ = (3 - k) / (-7 - 5k) = 1/7

Cross-multiplying:

7(3 - k) = 1(-7 - 5k)

Expanding and simplifying:

21 - 7k = -7 - 5k

Subtracting -5k from both sides:

21 - 7k - (-5k) = -7 - 5k - (-5k)

21 - 7k + 5k = -7

Combining like terms:

21 - 2k = -7

Adding 2k to both sides:

21 - 2k + 2k = -7 + 2k

21 = -7 + 2k

Adding 7 to both sides:

21 + 7 = -7 + 2k + 7

28 = 2k

Dividing both sides by 2:

28 / 2 = 2k / 2

14 = k

Therefore, k = 14.

Now that we know the value of k, we can find the midpoint of PQ:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Midpoint = ((8 + 1-5k) / 2, (k + 3) / 2)

Substituting k = 14:

Midpoint = ((8 + 1-5-14) / 2, (14 + 3) / 2)

Midpoint = ((8 - 69) / 2, (17) / 2)

Midpoint = (-30.5, 8.5)

Therefore, the midpoint of PQ is (-30.5, 8.5).